Even if David Roberts is right, I'll give an answer; more details my be found in the book

R. Brown, P.J. Higgins, R. Sivera, *Nonabelian algebraic
topology: filtered spaces, crossed complexes, cubical homotopy
groupoids*, EMS Tracts in Mathematics Vol. 15, 703 pages. (August
2011).

of which a pdf may be downloaded from

http://pages.bangor.ac.uk/~mas010/nonab-a-t.html

Any crossed module $\mu: M \to P$ has a classifying space $X=B(\mu)$ whose homotopy groups are 0 above dimension 2 and in dimensions 1 and 2 are respectively Coker $\mu$ and Ker $\mu$. A morphism $f$ of crossed modules is a weak equivalence iff the induced map of classifying spaces is a weak equivalence. The homology of a crossed module is defined as the homology of its classifying space. The result asked for is easily obtained by passing to universal covers.

For more information see also

{Ellis, Graham J.},
TITLE = {Homology of {$2$}-types},
JOURNAL = {J. London Math. Soc. (2)},
VOLUME = {46},
YEAR = {1992},
NUMBER = {1},
PAGES = {1--27},

and the subsequent correction.