Definition of Crossed homomorphism Suppose a group $G$ is acting on an abelian group $M$. Then a mapping $\phi: G \rightarrow M$ is called a crossed homomorphism if it satisfies the condition: $\phi(gh)=\phi(g)(g\cdot \phi(h))$ for every $g,h\in G$. My question is, how we will specify the action of $G$ is left or right in the definition of a crossed homomorphism from $G$ to $M$? I found these definitions:
If the action is left then we write $\phi(gh)=\phi(g)(g\cdot \phi(h))$. If the action is right, then we write  $\phi(gh)=(\phi(g)\cdot h)\phi(h)$.
My doubt is, if the action is right, Why not $\phi(gh)=\phi(g)(\phi(h)\cdot g)$ ? Can anyone please clear this concept of left or right?

Normally the easiest way to deal with such problems is to avoid ever considering right group actions. Suppose that we are given a group $G$ and a right $G$-module $M$ with action $\star$
Then we may make $M$ a left $G$-module by defining a left $G$-action via
$$g \cdot m = m \star g^{-1}$$
Edit: With the clarifying comments (thankyou @Derek Holt and my apologies to those who enjoy right actions) it seems useful to add futher explanation.
If $\phi : G \to M$ is a cocycle under this induced left action then the induced map $\psi : G \to M$ (remembering that we are first undergoing the automorphism $G \to G : g \mapsto g^{-1}$ ) satisfies the following condition
\begin{align*}
\psi(gh) = \phi((gh)^{-1}) &= \phi(h^{-1}g^{-1}) \\
&= h^{-1} \cdot \phi(g^{-1}) + \phi(h^{-1}) \\
&= \psi(g)\star h + \psi(h)
\end{align*}
which is precisely the cocycle condition we are claiming.
