This is a follow-up of this question, where the definition of a quasi-abelian crossed module was given. Namely, a crossed module $\partial\colon F\to G$ is quasi-abelian if the embedding $\partial_Z\hookrightarrow \partial$ of its center into it is a quasi-isomorphism.

**Question 1.** Assume that $\partial\colon F\to G$ is a crossed module which can be related
by a chain of quasi-isomorphisms to a quasi-abelian crossed module.
Then $\pi_1(\partial):={\rm coker}\ \partial$ is abelian and acts trivially on
$\pi_2(\partial):=\ker\partial$ (is the assertion about the action true?).
What else can be said about $\partial$?

**Question 2.** Assume that $\partial$ and $\partial'$ are two quasi-abelian crossed modules
which can be related by a chain of quasi-isomorphisms.
Is it true that their centers $\partial_Z$ and $\partial'_Z$ are isomorphic
in the derived category of complexes of abelian groups,
i.e. can be related by a chain of quasi-isomorphisms of complexes of abelian groups?

Both questions are motivated by cohomology with coefficients in crossed modules.