To put it simply they weren't told. Even Hirohito and the inner circle were not told of the complete truth until several days after the battle. And on a joint IJA/IJN meeting on the 10th the extent of the loss was concealed from the army and a cabinet minister was appointed to oversee control of the information. While the Combined Fleet did not return in full until June 14 in poor weather and fog.

The Naval General Staff then ordered from Hirohito that survivors and wounded would be returned under secrecy and returned to service ASAP with little to no contact to anyone not required including family. This included placing Kido Butai survivors in special security controlled wards for months. Those able to serve again were sent to either backwater posts or fed into the meat grinder of the South Pacific as fighting started in the Solomons. Even a civilian cameraman who had been aboard Akagi was held and warned not to return to Tokyo before being shipped out.

This also then represented the first time the Japanese govt outright fabricated news reports vs providing filtered rosy views of the war. Reports said the IJN had suffered two carries lost for two USN ones sunk. Even the Navy was fed missinformation, Kaga and Soryu we're taken off the ship registry list. Akagi and Hiryu were left on though and simply had no men assigned to them and said to be damaged.

For an excellent rundown you can pick up Shattered Sword by Parshall and Tully.