Basically, no. There's only one start and end date for the entire self enrolment method instance. So, that same start and end date pertain to all the keys in the same self-enrollment method, including any groups if you have that set. All the group key really does is match up the user to a group with the same key and put them in it. Otherwise, all the various settings including dates are shared by all the users of that same enrolment method instance in the course.
A possible workaround is to have multiple self-enrollment methods, each with a custom name and dates. You can have more than one, just give them different custom names. You could set up one method for each set of groups that share the same start & end date, eg. January groups, February groups. But if you have a lot of them that could get complex quickly.
A possible workaround is to have multiple self-enrollment methods, each with a custom name and dates. You can have more than one, just give them different custom names. You could set up one method for each set of groups that share the same start & end date, eg. January groups, February groups. But if you have a lot of them that could get complex quickly.