Answer 1
Cyrus the Great, Emperor of Persia, allowed the Jews to return from exile. He promised funds from the royal treasury for rebuilding the temple, although this work apparently did not commence in his lifetime. A successor, Darius I, implemented his policy at the end of the sixth century BCE (Ezra 4-6).
Answer 2
While Artaxerxes according to the book of Nehemiah released the Israelites from exile, it was more likely Cyrus. He is the leader that is given this attribution in Ezra 1, and he is also given the title of messiah or "anointed one" in Isaiah 45:1. Artaxerxes is highly unlikely.
King Cyrus allowed the Jews to return after the first exile of the Jews (586-516 BCE) in Babylon.
However, there have been numerous Jewish exiles. The Romans exiled the Jews from Judea in year 70 CE. The English exiled the Jews from England in 1290 CE. Spain exiled the Jews in 1492 and Portugal followed suit in 1497, creating a Jew-free Iberia. All of these different exiles were ended at different times.
Most kings and leaders of most countries did not allow the Jews to return from whatever exiles and banishments they were suffering from, either because they did not care, they were not in power over the states that had banished Jews, or they believed that the banishment was proper because Jews were evil in their eyes.
Cyrus the Persian gave the intial permission; and around 20 years later (after a hiatus), Artaxerxes (son of Ahaseurus, a.k.a. Xerxes) bade them continue building, and contributed material assistance.
It depends on where they were returning to and coming back from.
In 538 BCE the Persian king Cyrus permitted Jews to return to Judah.
Jesus did not rebuild the temple.
Cyrus the Great.
After defeating Babylon, Cyrus the Great of Persia allowed the Jews (and other people) to return to their homeland in 538 BCE.
A book of the bible did not allow anything, people did. If the question intends to ask: "In what book of the Bible does it discuss the return of the Jews from exile and the rebuilding of the Temple in Jerusalem?" there are two answers. The Book of Ezra and the Book of Nehemiah discuss this.
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.
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.
The return from exile predated Jesus by over five centuries.
Jews have returned to Jerusalem. However, there is no Jewish group seriously considering rebuilding the Temple as this would require destroying the Dome of the Rock Shrine of the Muslims and probably start a massive war.
King Solomon built the first Temple in 957 BCE with resources provided by his father King David. The second temple was allowed to be built by King Cyrus of Persia while the Jews were exiled in Babylon (536 BCE). Cyrus made a proclamation that the Temple in Jerusalem would be rebuilt, and the surviving Jews would return to rebuild it. This edict was supported by King Darius (522-486 BCE) who followed King Cyrus. This is described in the book of Ezra.
The Jews of the Babylonian Exile were granted the right to return to the Land of Israel, and rebuild the Temple which had been destroyed by Nebuhadnetzer.
The Persian king Cyrus after he defeated the Babylonians, released the Israelites who were taken prisoners by the Assyrians He allowed them to take the temple vessels and helped rebuild the destroyed temple..
They allowed the Hebrews to rebuild their Temple in Jerusalem and to have general religious freedom and autonomy within the Persian Empire.