No one knows. Period.
Normative Judaism does not plan or push for the rebuilding of the Temple. We pray to God for His redemption and all that will be included with it, and that's all. One of the purposes of the present exile was/is to teach us the quality of humility and patience.
(There are tiny fringe cliques who attempt to take some action or another towards the rebuilding of the Temple, but these people do not have the approval of any major Rabbinic figure. And their caprices serve to potentially endanger the Jewish people. Not to mention that we don't today have the status of ritual purity which would theoretically enable us to deal with matters concerning the Temple.)
The return of several tens of thousands of Jews from the Babylonian exile and the rebuilding of Jerusalem and the Temple.
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.
The Persian King Cyrus the Great allowed the Jews to return to Jerusalem and rebuild their temple. After conquering Babylon in 539 BCE, he issued a decree permitting exiled Jews to return to their homeland and restore their religious practices. This event is documented in the biblical books of Ezra and Isaiah, highlighting Cyrus's role in facilitating the Jewish return and the rebuilding of the Second Temple.
Orthodox Jews believe that the Temple will be rebuilt.
That was the Bar Kokhba revolt (132-136). The Jews revolted against the emperor Hadrian renaming Jerusalem with the Roman name of Aelia Capitolina rebuilding Jerusalem as a Roman town, building a Roman temple in the place of the destroyed Second Temple and forbidding circumcision. When they were defeated Hadrian persecuted Judaism and renamed Judea Syria Palaestina (after the Philistines) to erase the memory of Judea, as well forbidding the Jews from entering Jerusalem.
In the book of Ezra, the enemies of the Jews who were rebuilding the temple in Jerusalem lodged a complaint against them with King Artaxerxes. They accused the Jews of planning to rebel and rebuild a city that would no longer pay tribute to the king. This led to a temporary halt in the temple's reconstruction until the matter was addressed.
If you are referring to the story of Hanukkah, a statue of Zeus was erected inside the Holy Temple in Jerusalem.
They destroyed it.
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.
The Jews built the Temple under the authority of King Solomon.
Both Cyrus and Darius gave instructions, materials and encouragement towards the rebuilding of the Temple (Ezra ch.6). However, the actual construction was done by the Jews. Concerning the rebuilding of Jerusalem itself, Artaxerxes gave permission to rebuild it (Nehemiah ch.2); but here too, it was the Jews who did the construction.
This did not happen. The Second Temple stood while Jesus was alive and remained standing for four decades after Jesus was killed by the Romans. The Third Temple has not been built.Jesus was never in exile, didn't return from one, and was never involved in any way in rebuilding the Temple in Jerusalem. Jesus came to earth approximately 536 years after the return of the Jews to rebuild the temple.An observation:Just in case the question meant to read "Jews" instead of "Jesus" (which would then completely make sense), the answer would be King Cyrus of Persia (Ezra 5:13). Zerrubabel led the Jews, along with Ezra and Nehemiah in the rebuilding of the Temple.