Greetings! This is indeed a question which has been something of a thorn in the sides of First World War historians, as if the Serbian government was indeed involved in the assassination plot, then the historiographical debate has an entirely new avenue to explore. For the most part, historians in the previous decades have generally avoided making any definitive statements on the connection between the Ujedinjenje ili smrt (Unification or Death - the proper name for the Black Hand Gang) society and the Serbian Government, for lack of any primary sources on the matter. They instead focus on the Serbian government's struggle with South Slav unity and the antagonism of the Habsburgs in the process. So let's see if we can shed some light (even if it required some speculation) on this query:

Firstly, a quote from Michael Howard in his Very Short Introduction to the First World War, which I think encapsulates the overriding idea of the Black Hand's connection to the Serbian government:

"Then on 28 June 1914 the heir to the Habsburg throne, Archduke Franz Ferdinand, was assassinated in Sarajevo, the capital of Bosnia-Herzegovina, by Gravil Princip, a teenage terrorist trained and armed by the Serb-sponsored Black Hand."^(1)

That's pretty much all Howard mentions of the Black Hand Gang, and most historians echo this lack of focus on the background of the Black Hand. What we do know concretely is that the Black Hand Gang was essentially the terrorist offshoot organisation of the Narodna Odbrana (National Defense) society, founded in 1908 just after the Austro-Hungarian annexation of Bosnia and Herzegovinia, by Serbian ministers and army personnel.^(2) However, even there we have a few problems, because historians are still debating as to whether event the Black Hand leadership was aware of the plot to assassinate an Austrian archduke, or whether Princip was acting out of his own fanaticism to the cause. Vladimir Dedijer, a pre-eminent writer on the Sarajevo assassination, on the complexities of this topic:

"One of the most controversial issues of modern history arises from this question: What were Princip's motives and who were his instigators, if any, and his accomplices? Sir Edward Grey [British Foreign Secretary in 1914] described Princip's crime as the perfect political murder, in the sense that it would be impossible for the truth ever to be established.^(3)"

Dedijer and his fellow historians focus in particular on the connection between Princip and one man: Dragutin Dimitrijević, otherwise known as "Apis". A founding member of the Black Hand Gang and one of its executive leaders, Apis had taken part in the 1903 murder of King Alexander and by 1912, was Chief of Serbian military intelligence.^(4) Therein lies the crucial factor: if Apis knew and sanctioned the assassination plot, then it would reveal much about the connection between this terrorist group and the Serbian government. However, that's as far as the concrete evidence goes, and anything about persons higher than Apis knowing of the plot is fairly hard to come by (and speculative, for the most part). Harold Temperley on the matter:

"It seems therefore that the Serbian government must be acquitted of any connection with the Black Hand, or of any direct complicity in the plot. The thing of which they can perhaps be accused of is that of having known something of the plot beforehand and of having failed to warn the Hapsburg Government."^(5)

Note how conditional Temperley's remarks are: we probably will never know for sure if the Serbian government even had concrete knowledge on the plot to assassinate Franz Ferdinand, or if they merely knew about extremist actions being carried out in the area. It might also interest you to know that immediately after the war broke out, scholars and government officials alike were actually accusing greater powers of instigating the assassination attempt. These included the Russian General Staff, known for its pan-Slavic sympathies and strong ties with the Balkan region, and even the German General Staff, whom some conspiracy purveyors believed motivated Apis to order the attempt in order to further their own goals in Europe. These theories are even more unfathomable at face value, and even looking at the historiography on them is enough to come to a similar conclusion as Dedijer did:

"The theories that both Princip's group and Colonel Apis were moved to action by some outside force (the Russian General Staff or the German General Staff, as some historians claim) seem to be based on inadequate evidence.^(6)

With all these theories and persons being so critical to the analysis of that fateful shot on the 28th of June, 1914, it seems almost perplexing that historians have yet to put the matter to rest. Dedijer himself wrote an entire publication on the subject, titled The Road to Sarajevo (1966) and even then the answer is less conclusive than one might have hoped. Laurence Lafore, in reviewing his book, sums the current response to your question rather neatly:

"The answer to the explosive question, argued so long and passionately, "Did the Serbian cabinet know of the plot?" turns out to be, "Yes and no." Some members knew something, and they tried, not very resolutely or efficiently, to stop it."^(7)

On that note, I would highly encourage that you pursue further reading on the matter by perusing the source articles below, which detail far more the methodology of historians in coming to this conclusion. Grabbing a copy of Dedijer's book would be brilliant as well, but seeing as it's out of print and rather expensive online, these articles should suffice. I hope you found this response enlightening and any follow-ups on the specifics, let me know!

Sources:

[1]: Howard, Michael Eliot. The First World War: A Very Short Introduction. Oxford: Oxford University Press, 2007.

[2]: Temperley, Harold. "The Coming of the War." Foreign Affairs 9, no. 2 (1931): 317-38. Accessed January 11, 2021. doi:10.2307/20030352. Accessible with JSTOR here.

[3]: Dedijer, Vladimir. "Sarajevo Fifty Years After." Foreign Affairs 42, no. 4 (1964): 569-84. Accessed January 11, 2021. doi:10.2307/20029714. Accessible with JSTOR here.

[4]: Williamson, Samuel R. "The Origins of World War I." The Journal of Interdisciplinary History 18, no. 4 (1988): 795-818. Accessed January 11, 2021. doi:10.2307/204825. Accessible with JSTOR here.

[5]: Temperley, Harold. "The Coming of the War."

[6]: Dedijer, Vladimir. "Sarajevo Fifty Years After."

[7]: Lafore, Laurence. The Journal of Modern History 40, no. 3 (1968): 439-42. Accessed January 11, 2021. http://www.jstor.org/stable/1878169.