Persian Empire.
They allowed the Hebrews to rebuild their Temple in Jerusalem and to have general religious freedom and autonomy within the Persian Empire.
Cyrus the Great.
To go back to their homeland and rebuild the Holy Temple.
About 40,000 of the Hebrews aristocracy who had been deported to Babylon and were willing to return and try to recover land they claimed in Judah.
This is a sentence, not a question. If you meant to ask "Did the Israelites return and rebuild the temple?" or you meant to ask "Will the Israelites return and rebuild the temple?", see the two Related Questions below.
In the Temple in Jerusalem.
David made Israel a mighty empire, , and Jerusalem as the capital city, and david started building the temple, but it was Solomon who finally finished building the temple of God.
There was no 'church' and Israel was in northern Palestine. Hebrews who had been deported to Babylon by the Babylonians were allowed to return to Jerusalem to attempt to reclaim ancestral lands, and on the side rebuilt the destroyed temple.
Jesus did not rebuild the temple.
Ezra was to rebuild the destroyed temple.
Yes. In 539 BCE, King Cyrus the Great of Persia gave permission for the Jews of Babylon to return to their ancestral homeland and rebuild the Temple.
It is very unlikely that the government of Israel will ever rebuild the temple. There is a Muslim holy site on the Temple Mount. Furthermore, the Israeli government is not a religious government. Moreover, Orthodox Jews believe the Temple cannot be rebuilt until the messiah comes.