  A quandle is a set that has a binary operation satisfying three conditions
corresponding to the Reidemeister moves. Homology theories of quandles have
been developed in a way similar to group homology, and have been applied to
knots and knotted surfaces. In this paper, a homology theory is defined that
unifies group and quandle homology theories. A quandle that is a union of
groups with the operation restricting to conjugation on each group component is
called a multiple conjugation quandle (MCQ, defined rigorously within). In this
definition, compatibilities between the group and quandle operations are
imposed which are motivated by considerations on colorings of handlebody-links.
A homology theory defined here for MCQs take into consideration both group and
quandle operations, as well as their compatibility. The first homology group is
characterized, and the notion of extensions by $2$-cocycles is provided.
Degenerate subcomplexes are defined in relation to simplicial decompositions of
prismatic (products of simplices) complexes and group inverses. Cocycle
invariants are also defined for handlebody-links.
