When did this rebuilding begin?

Ezra 5:2 (ESV)

2 Then Zerubbabel the son of Shealtiel and Jeshua the son of Jozadak arose and began to rebuild the house of God that is in Jerusalem, and the prophets of God were with them, supporting them.

The time is August, September, and October 520 BC. The Persian Empire has just gone through a period of internal strife and is enjoying a time of relative peace. This is the perfect time to start a building project on the outskirts of the Persian kingdom. Through the prophets Haggai and Zechariah, God stirs up Zerubbabel and Joshua to begin work on rebuilding the temple.