Currently we have tags for ring-theory, group-theory and semigroups and separate tags for ring-homomorphism, group-homomorphism and semigroup-homomorphism. (The latter tag is used a total of 11 times, 6 times by the same user.)
In my opinion, the tags about the morphisms add little value to a question and should be made synonyms of the bigger tags.
While it is true that for example, not every group theory question focuses on the properties of group homomorphisms, it's difficult to find much group theory questions that don't involve group homomorphisms in some way. The set of users who are interested in reading or answering questions about group homomorphisms is almost surely identical with those that are interested in group theory.
If someone has a specific question about properties of group homomorphisms, then this can be made clear in the title and question text. I also don't think that the tag on group homomorphisms makes questions that are specifically about group homomorphisms easier to find, since it is difficult to ensure that it is consequently used on such questions (Currently, there are many questions about group homomorphisms, which only have the group-theory tag)
I could of course post this in the tag managment thread, but I feel like this is a general principle that should be discussed. If there's a consensus that seperate tags for morphisms are worthwhile, then I think it would be only consequential to create tags such as monoid homomorphisms and module homomorphisms etc.
Note that ring-morphisms is already a synonym for ring-theory and the proposal to make ring-homomorphism a synonym of ring-theory is pending with 3 upvotes. I have also proposed group-homomorphism as a synonym for group-theory. (I don't have the score to propose semigroup-homomorphism as a synonym to semigroups)