According to Wikipedia (and to many other sources):
The first cohomology group is the quotient of the so-called crossed homomorphisms, i.e. maps (of sets) $f : G \to M$ satisfying $f(ab)=f(a)+af(b)$ for all $a$, $b$ in G, modulo the so-called principal crossed homomorphisms, i.e. maps $f : G \to M$ given by $f(a) = am−m$ for some fixed $m\in M$.
Coboundaries are generally such cocycles which "doesn't matter", that is, adding them to a cocycle doesn't change some property of the cocycle. For example, in group cohomology, 2-coboundaries doesn't change the group extension defined by a 2-cocycle (see here).
My question: what is the property of a crossed homomorphism that remains unchanged by adding a principal crossed homomorphism to it?
The closest to my wish is May's Proposition 4.3:
Proposition 4.3. The groups $Z^1(G;M)$ and $Aut(G;M)$ are isomorphic, and the isomorphism restricts to an isomorphism $B^1(G;M)\cong Inn(G;M)$. Therefore $H^1(G;M)\cong Out(G;M)$.
Here $Out(G;M):=Aut(G;M)/Inn(G;M)$, and $Aut(G;M)$ is the automorphisms group of the canonical split extension of $M$ by $G$, and $Inn(G;M)$ is the inner automorphism group of it, that is the group of conjugations by the elements of $G$.
So, if we call two automorphisms essentially the same, if they differ by a conjugation, and we identify the crossed homomorphisms with $Aut(G;M)$ and the principal crossed homomorphisms to $Inn(G;M)$, then we can say that an a principal crossed homomorphism essentially doesn't change the crossed homomorphism. But this seems a bit far-fetched to me.