Charles Rezk had highlighted in MO:q/123624 that "Nearby homomorphisms from compact Lie groups are conjugate", and in consequence -- further highlighted in Remark 2.2.1 of his Global Homotopy Theory and Cohesion -- that for $G$ and $\Gamma$ Lie groups with $G$ compact, the homomorphism space $\mathrm{Hom}(G,\Gamma) \subset \mathrm{Maps}(G,\,\Gamma)$ decomposes under the adjoint action of $\Gamma$ as:

$$ \mathrm{Hom}(G,\,\Gamma) \;\;\;\;\simeq\; \underset{ { [\phi] \in } \atop { H^1_{\mathrm{Grp}}(G,\,\Gamma) } }{\coprod} \; \Gamma/\mathrm{Stab}_\Gamma(\phi) \;\;\;\;\; \in \Gamma \mathrm{Act}(\mathrm{TopSp}) \,. $$

I am wondering: Is the generalization of this statement to crossed homomorphisms still true?

To recall, for $G$ and $\Gamma$ two groups, and in addition for

$$ \alpha \;\colon\; G \longrightarrow \mathrm{Aut}_{\mathrm{Grp}}(\Gamma) $$

an action of $G$ on $\Gamma$ by group automorphisms, a function $\phi \;\colon\; G \longrightarrow \Gamma$ is a crossed homomorphism if

$$ \underset{g_1, g_2 \in G}{\forall} \;\;\; \phi(g_1 \cdot g_2) \;=\; \phi(g_1) \cdot \alpha(g_1) \big( \phi(g_2) \big) \,. $$

Moreover, a pair $\phi$, $\phi'$ of such crossed homomorphisms is "crossed conjugate" (this terminology I seem to making up, but the definition is standard) if there exists $\gamma \in \Gamma$ such that

$$ \underset{g \in G}{\forall} \;\;\; \phi'(g) \;=\; \gamma^{-1} \cdot \phi(g) \cdot \alpha(g)(\gamma) \,. $$

So I am wondering: Are nearby crossed homomorphisms crossed conjugate?

I like to think about this as follows (by this Prop.):

The crossed-conjugation groupoid of crossed homomorphisms is isomorphic to the groupoid of (strict, i.e. $\mathrm{Grpd}$-enriched) sections of the groupoid bundle $\mathbf{B} (\Gamma \rtimes_\alpha G) \xrightarrow{ \mathbf{B}\mathrm{pr}_2 } \mathbf{B} \Gamma$ which is the groupoidal delooping of the semidirect product group projection.

In other words:

A crossed homomorphism $\phi \,\colon\, G \xrightarrow{\;} \Gamma$ is the same as an actual homomorphism of the form $\big( \phi(-),\, (-)\big) \;\colon\; G \longrightarrow \Gamma \rtimes G$; and under this identification, a crossed conjugation is just a plain conjugation with elements of the form $(\gamma, \mathrm{e})$ in $\Gamma \hookrightarrow \Gamma \rtimes G$.

In this equivalent reformulation, the question becomes:

For $G$ and $\Gamma$ Lie groups, with $G$ compact, are nearby homomorphisms of the form $\big( \phi(-),\, (-) \big) \;\colon\; G \xrightarrow{\;} \Gamma \rtimes G$ conjugate not just by any element in $\Gamma \rtimes G$, but by an element of the form $(\gamma, \mathrm{e})$ in $\Gamma \hookrightarrow \Gamma \rtimes G$?

Using just the fact that nearby plain homomorphisms out of compact Lie groups are plain conjugate, one can immediately deduce that this crossed generalization is true in the special case that the action $\alpha \,\colon\, G \to \Gamma \rtimes G$ restricts to the trivial action on the center of $G$. However, this special case does not seem to be too interesting. For example in the archetypical application to twistings of Real K-theory, $G$ is abelian.

So it looks like to make progress here one needs to go into any one of the proofs that construct the conjugation of nearby homomorphisms and see if they can be enhanced to incorporate the above "section constraint" in the case that the codomain group is a semidirect product with the domain group. I admit that I haven't really looked into these proofs yet, firing off this question first to see if anyone has thought about it.