If $H \lhd G$, then $G$ acts on $H$ by conjugation. I need to talk about this action but in a situation where $H$ is not (necessarily) normal. When $H \leq G$, there is a "partial action" of $G$ on $H$ by $G$-conjugation, and clearly it has very nice properties, for example $h^g h^{g'} = h^{gg'}$ and $(hk)^g = h^g k^g$ when these expressions are defined. To my amazement (at how bad I am at searching), I did not find a discussion of this anywhere. I'm sure it is very standard, and I'm looking for some pointers for a good formal setting.

Question. Is there a good basic reference for the "partial conjugation action" of a group on a subgroup, in some formalism? Is there a standard way to talk about this object?

I know many ways to talk about partial actions, and I can specialize them to my case, so in theory I can solve this question in many ways myself. However, it seems like such a natural example of a partial action that the fact I am not finding this discussed anywhere suggests to me that I may be missing something, so a reference or situation where this appears would be nice.

A more serious problem that tripped me up is "generation". In many of the settings of partial actions, it is difficult to state that a partial group action is finitely-generated. If $G = \langle S \rangle$, then the partial actions given by partial conjugation by $s \in S$ on a subgroup $H \leq G$ obviously "generate" the partial action of $G$ in some sense. But I don't really know how to say this in a good way, especially I run into issues with domains, when trying to state write down the axioms, and I don't want to reinvent the wheel.

More specifically, it is natural enough to say that a group action $G \curvearrowright H$ is finitely generated when $G$ is, so...

Question. Is there a standard way to say a "partial group action" is "finitely-generated", so that in the case above of a finitely-generated group $G$ partially acting on its subgroup $H$ by conjugation, the partial action of $G$ would indeed be finitely-generated?

I tried to look at some existing formalisms for partial actions. One thing you can do is form the action groupoid of $G$ acting on itself by conjugation, so take the action groupoid with $\Gamma = G$ acting group, $\Omega = G$ the set where it acts, then take the subgroupoid for the restriction $H \subset \Omega$. Unfortunately then it is a bit awkward (I think) to discuss individual partial bijections that may or may not end up being part of such an action (they should of course be "partial automorphisms", but the list of axioms does not seem to be suggested by the subgroupoid definition). It is also not obvious to me how to state finite generation correctly. Groupoidification drops the group, and after taking the subgroupoid corresponding to $H$, it might not even be determined up to isomorphism, so when you say that a groupoid of partial automorphisms on $H$ is finitely-generated, there is no $G$ this could possibly refer to. Furthermore a "finitely-generated groupoid" seems to usually have a finite number of objects, so this doesn't look correct.

I then thought of pseudogroups, but all pseudogroup references I found deal with pseudogroups of homeomorphisms, and discuss mostly orthogonal issues. By any definition of a pseudogroup I could find, an action by partial automorphisms is not really a pseudogroup (unless I introduce some topological structure that I'm not going to use). Furthermore, I did not find a discussion of finite generation that tells me what I should do with domains.

There is also literally the notion of a partial action of a group. I thought of this last because I had never actually seen this before (people I know only talk about groupoids), but this was maybe the most promising formalism. I guess I would like to discuss the representations of groups by the "inverse semigroup of partial automorphisms of a group", but I don't know the jargon, and at least based on a brief look I did not find a notion of finite generation with the correct properties.