  This paper links the third symmetric cohomology (introduced by Staic and
Zarelua ) to crossed modules with certain properties. The equivalent result in
the language of 2-groups states that an extension of 2-groups corresponds to an
element of $HS^3$ iff it possesses a section which preserves inverses in the
2-categorical sense. This ties in with Staic's (and Zarelua's) result regarding
$HS^2$ and abelian extensions of groups.
