My knowledge of the Pacific War doesn't include expertise on the exact technical details, but radio transmissions could usually be sent well after invasions had begun.

In fact, a team sent to Iwo Jima to repatriate the remains of Japanese soldiers relatively recently found a large radio transmitter inside one of the caves believed to be part of the command structure. The invasion began on February 19, 1945 after a lengthy naval bombardment (that was still insufficient) and the famous photo of the flag raising was taken on Mount Suribachi on February 23, but the battle continued long after that. According to one account, a song dedicated to the defenders of Iwo Jima was broadcast from Tokyo on March 14. General Tadamichi Kuribayashi, still alive and leading what was left of his men, responded with a message of thanks. Kuribayashi was able to send and receive message for most of the battle, providing at least some account of what was happening on the ground. Communication on the island itself still would have been difficult, though. Every cave, tunnel, pillbox, and bunker couldn't be interconnected or have a radio. Some units would be cut off from communication.

Kuribayashi's letters home (some of which are in here) show that he always expected to die on Iwo Jima, even months before the invasion began. But some of his radio transmissions showed how disillusioned he was during the battle. He knew his men were doomed to die not because they were poor fighters, but because there was no way to overcome the American fleet and aircraft that isolated his garrison. The final messages from Iwo Jima reportedly came through to Chichi Jima on March 24. According to Samuel Eliot Morrison:

General Kuribayashi's radio informed Major Horie at Chichi Jima on the 21st: "We have not eaten or drunk for five days. But our fighting spirit is still high. We are going to fight bravely till the last." On the 24th Horie received the last word from his commanding general: "All officers and men of Chichi Jima, goodbye."

The precise setup would have varied from battle to battle, and again, is beyond my level of expertise. However, I can give you one more example.

The defense of Iwo Jima in February/March 1945 was designed around caves and tunnels (man-made, natural, and modified natural features). It bore many similarities to the Battle of Peleliu in fall 1944. I'm unaware of whether Kuribayashi's strategy was directly influenced by the Battle of Peleliu; he had been named commander of the garrison at Iwo Jima before Peleliu was invaded.

According to archaeologists Neil Price and Rick Knecht, who visited Peleliu and published a paper in 2012 a few years ago, an underwater cable kept the commander at Peleliu in constant communication with the overall Japanese command of the Palaus about 30 miles north:

Command of Peleliu’s defence was given to Col. Nakagawa Kunio, in direct radio communication with headquarters on Koror — and thus Japan — by means of a submarine cable (a link that remained operative and undetected by the Americans throughout the battle).

I'm not sure that setup would have worked at Iwo Jima, where the next closest Japanese base of any substance (Chichi Jima) was more than 100 miles away, but you get the idea.

Elsewhere in the same paper:

By way of encouragement, the Peleliu garrison were made aware by radio that the whole of Japan was expectantly following their sacrifice. No less than eleven telegrams were sent to the defenders by the Emperor himself during the course of the battle, a number unequalled in any other conflict of the war (Harries and Harries, 1991: 360).

I've not seen details about communication with other islands during those battles, though I've never really gone looking. However, it seems that in most cases, there was at least some level of radio communication available to Japanese commanders at least in the early stages of battle.