# Action on identity in crossed modules

Crossed modules are given by the data $$(H \xrightarrow{t} G,\alpha: G \times H \to H)$$ where $$\alpha$$ is an action and $$t$$ is a group homomorphism between a pair of groups, satisfying two conditions: equivariance and Pieffer identity. See the full definition here. We will denote $$\alpha(g,h) = g\cdot h$$.

In all the examples listed in the wikipedia page, the following equation is satisfied: $$g \cdot e_H = e_H,$$ where $$e_H$$ is the identity of $$H$$. It is easy to show this if $$t$$ is an injection or a surjection. I don't know if this statement is true in general.

So the question is:

Prove or disprove: $$g \cdot e_H = e_H$$ for any $$g \in G$$ in a crossed module.

Judging by the definition of wikipedia, if $$G$$ acts on $$H$$ by automorphisms this means that for any $$g\in G$$ the function $$g\cdot -:H\rightarrow H$$ is a group-automorphism. In particular it is a group-homomorphisms, sending the identity element $$e_H\in H$$ to itself. In other words $$g\cdot e_H = e_H$$ for any $$g\in G$$.