Let $C$ be a small poset category. I'm coming across a category where objects are (certain) functors valued in groups $F\colon C\to\operatorname{Grp}$, and morphisms between two objects $F$ and $G$
are *almost* natural transformations $\eta\colon F \Rightarrow G$.
The difference is that if $f\colon e \to v$ is an arrow of $C$, we are allowed an inner automorphism $\Phi\_e\colon Gv \to Gv$ such that the following diagram commutes
$$\require{AMScd}\begin{CD} Fe @>\eta\_e>> Ge \\
@VVFfV @VV\Phi\_eGfV \\ Fv @>\eta\_v>> Gv.\end{CD}$$


I'm curious if this "naturality up to automorphism" condition occurs elsewhere in the wild and if you know of a name for it?



Naturality up to (inner) automorphism?
Assuming the inner automorphisms you get are assumed to satisfy some further coherency conditions, your morphisms should amount to [pseudonatural transformations](https://ncatlab.org/nlab/show/pseudonatural+transformation). This requires thinking of the category of groups as a 2-category.


