Well, the most common names in the actual Achaemenid dynasty are pretty easy to pick out. As given names: Arsames (5), Darius (4), and Hystaspes (3) take the medals (Arshama, Darayava^(h)ush, and Vishtaspa, in Old Persian if you want to get technical). The popularity of all three is further reinforced by a large number of nobles with those names apparently outside of the royal family.

This contrasts with the names the kings are usually known by, where Artaxerxes is undoubtedly the top contender, with 5 kings/claimants, but all but Artaxerxes I adopted it as a regnal named. The same is actually true for Darius, where Darius II and III both adopted the name after coming to power, while three of the four royals named Darius at birth died as princes. However, in the post-Achaemenid period, both Artaxerxes and Darius (which became Middle Persian Ardashir and Dara), became very common noble names.

Cyrus is actually a surprisingly uncommon name in Achaemenid history, with just Cyrus the Great, his grandfather (Cyrus I), and Cyrus the Younger. For whatever reason, there do not seem to have been many Persians named for the Empire's founder. Some scholars speculate that this was a side effect of general suppression of the Teispid family (Cyrus' house if you take the view that Darius the Great was totally unrelated rather than a distant cousin).

Several women's names are also identifiable as prolific in the royal family. Amestris, Rhodugne, and Roxane can each claim three Achaemenid noble women. A somewhat unisex name: Ochus/Ocha also covers three royals: the birthname of both Darius II and Artaxerxes III, as well as Artaxerxes III's mother-in-law/half-sister. Interestingly, Ochus is just the Latin rendering of Oxos, as in the ancient name for the Amu Darya river.

Outside of the royal family, variations of Megabyzus and Mithridates were both popular, exceedingly so for the latter. Unlike most of the common royal names, except for Artaxerxes, these are both theophoric names (ie named for gods). In Persian names, Greek Mega- is usually understood as poor translation of Persian baga, literally "god," and Mithridates obvious pays homage to the god, Mithra. Use of "Mithridates," spikes dramatically in sources after the late-5th Century BCE, corresponding precisely with the growing importance of Mithra under Darius II and his successors. Other names appear frequently in noble families

So to answer the specific version of your question:

Were names like Cyrus, Darius and Xerxes common names that one could find inside the iranians of the empire.

No, Cyrus and Xerxes weren't even all that common in Persia so far as we can tell. Darius does seem to have been relatively common but isolated to the province of Parsa, and may actually be more common after the Achaemenid period. However, to answer the broader version:

Did different Iranian groups have different sets of names

This is a mixed bag. We don't have great indicators for which people had which Iranian ethnic backgrounds in the Achaemenid period. Greek sources tend to identify any Iranian leader as a Persian, and Persian sources only rarely identify specific individuals' ethnicity. As a result, most of our ethnically specific names are one offs or associated with a specific dynasty (mostly Medes).

Looking immediately after the Achaemenid period, to the Hellenistic and early Parthian eras, there were absolutely names that were either more common or specific to one Iranian ethnicity. Darius/Dara or Artaxerxes/Ardashir in Persia or Volgasses/Volaxsh and Phraates/Frahat for Parthian. (Interesting to note the popularity of Phraates in Parthian might be one sign that what we call Parthian is actually the Middle Iranian form of Old Median, eg Phraortes/Fravartish).

However, across both the Old and Middle periods of the Iranian languages, the overwhelming majority of examples are religious names, often just straight up the names of gods like Hormazd, Bahram, or Mihr. Cultural heroes, like Hystaspes, Atossa, Kavad, and Khosrow, were also fairly popular. Names containing elements like Arta-, Baga-, Yaz-, or Fra- were perpetually common across the Iranian world before Islam because of the region's shared Zoroastrian heritage.

If you want to go deeper on all of the known ancient Persian names, and happen to know German, the Iranischem Namenbuch by Ferdinand Justis is an 1895 attempt to catalog every Achaemenid name known at the time, and has been expanded on over the last 127 years by publications in Iranische Onomastik under the title Iranisches Personnennanamenbuch. If you're German isn't great or you don't want to deal with the considerable time and/or financial investment to track down the various editions of the Namenbuch Avest.org has a sizable list of ancient Iranian names as well.