Compatible with trivial action iff image centralizes inner automorphisms

Statement
Suppose $$G$$ and $$H$$ are groups with homomorphisms $$\alpha:G \to \operatorname{Aut}(H)$$ and $$\beta:H \to \operatorname{Aut}(G)$$ such that $$\alpha$$ is the trivial homomorphism, i.e., it sends every element of $$G$$ to the identity automorphism of $$H$$.

Then, the following are equivalent:


 * 1) $$\alpha$$ and $$\beta$$ form a  compatible pair of actions.
 * 2) The image of $$\beta$$ in $$\operatorname{Aut}(G)$$ lies inside the centralizer $$C_{\operatorname{Aut}(G)}(\operatorname{Inn}(G))$$, i.e., it commutes with all the inner automorphisms of $$G$$.

Related facts

 * Trivial pair of actions is compatible

Proof
The actions are compatible if and only if the following two conditions hold, where $$\cdot$$ denotes the action $$\alpha,\beta$$ or conjugation within a group, as is clear from context:

$$g_1 \cdot (h \cdot g_2) = (g_1 \cdot h) \cdot (g_1 \cdot g_2) \ \forall \ g_1,g_2 \in G, h \in H$$

$$h_1 \cdot (g \cdot h_2) = (h_1 \cdot g) \cdot (h_1 \cdot h_2) \ \forall g \in G, h_1,h_2 \in H$$

Using the triviality of $$\alpha$$, these conditions are equivalent to:

$$g_1 \cdot (h \cdot g_2) = h \cdot (g_1 \cdot g_2) \ \forall \ g_1,g_2 \in G, h \in H$$

$$h_1 \cdot h_2 = h_1 \cdot h_2 \ \forall g \in G, h_1,h_2 \in H$$

Note that the second condition is vacuously true, so it conveys no information. The first condition is equivalent to saying that the action of $$h$$ commutes with conjugation by $$g_1$$, which is equivalent to saying that $$\beta(h) \in C_{\operatorname{Aut}(G)}(\operatorname{Inn}(G))$$.