It is permissible.
He should perform the qaḍā of prayers (i.e. perform them again later as compensation) that he has performed completely.
It depends on the intention of the wife, if she hopes to convince her husband to return then it is not considered as abandoning the homeland and if there is no hope, then it is considered to be an inevitable abandonment.
The time of the morning nāfilah starts before the Fajr prayer , f r o m the Fajr adhan until the appearance of redness in the east, and it is possible to perform that immediately after the night prayer.
There are some limitations for women and sometimes for men for entering mosques; however, there are much fewer limitations for entering hussainiyyahs. Of course one will be granted a much greater reward for performing prayer in the mosque compared to a husainiyyah and this is why both kinds of building are constructed.
