What is the reason Lot’s wife – upon him be peace – was considered among "those who remained behind" and punished along with her people, even though the abomination of sodomy for which they were punished specifically concerned males?

Lot's wife suffered the same fate as her people because she was an unbeliever and spied on Lot for the disbelievers. Allah has presented her and Noah's wife as an example for those who disbelieve, and the reason for her destruction was her disbelief and her support for her people. Interpreters of the Qur'an attributed the treachery mentioned in the verse: "{Allah presents an example of those who disbelieved: the wife of Noah and the wife of Lot. They were under two of Our righteous servants, but they betrayed them; so those [husbands] did not avail them from Allah at all, and it was said, 'Enter the Fire with those who enter.'}" (At-Tahrim:10), to betrayal of religion, not indecency; because the wives of prophets are immune from indecency.