  The purpose of this note is to observe that a homomorphism of discrete groups
$f:\Gamma\to G$ arises as the induced map $\pi_0(\mathfrak{M})\to
\pi_0(\mathfrak{X})$ on path components of some closed normal inclusion of
topological groups $\mathfrak{M}\subseteq \mathfrak{X},$ if and only if the map
$f$ can be equipped with a crossed module structure. In that case an
essentially unique realization $\mathfrak{M}\subseteq \mathfrak{X}$ exists by
homotopically discrete topological groups.
