Irreducible $G$-group, Irreducible $G$-module. Let $G$ and $M$ be two groups. Let $M$ be a $G$-group. What does an irreducible $G$-group mean? I have heard the term irreducibility for modules. So, what actually irreducibility means here? Is that something related to $G$-module? However, when we say $M$ is a $G$-module, in that case, $M$ has to be an abelian group. Here, $M$ is an arbitrary group. It does not make sense to me. Please help me with this.
For the definition of $G$-groups, you can refer to https://en.wikipedia.org/wiki/Group_with_operators.
For the definition of $G$-modules, you can refer to https://en.wikipedia.org/wiki/G-module.

In the paper
Lafuente, Julio P.; Lizasoain, I.; Ochoa, Gustavo, Projective G-groups, J. Pure Appl. Algebra 165, No. 2, 213-225 (2001). ZBL1017.20019,
an irreducible $G$-group is defined to be a group with an action of $G$ by group automorphisms that has no proper nontrivial $G$-invariant normal subgroups.
That doesn't prove that this is the same as the  definition intended in the paper
Detomi, E.; Lucchini, A., Crowns and factorization of the probabilistic zeta function of a finite group., J. Algebra 265, No. 2, 651-668 (2003). ZBL1072.20031,
where the OP said (in comments) that he encountered the term, but that paper refers to another paper co-authored by Lafuente for other terminology regarding $G$-groups, so it seems very likely that this is the intended definition.
