Internal crossed modules and Peiffer condition

In this paper we show that in a homological category in the sense of F. Borceux and D. Bourn, the notion of an internal precrossed module corresponding to a star-multiplicative graph, in the sense of G. Janelidze, can be obtained by directly internalizing the usual axioms of a crossed module, via equivariance. We then exhibit some sufficient conditions on a homological category under which this notion coincides with the notion of an internal crossed module due to Janelidze. We show that this is the case for any category of distributive $\Omega_2$-groups, in particular for the categories of groups with operations in the sense of G. Orzech.