  Biquandles and multiple conjugation biquandles are algebras which are related
to links and handlebody-links in $3$-space. Cocycles of them can be used to
construct state-sum type invariants of links and handlebody-links. In this
paper we discuss cocycles of a certain class of biquandles and multiple
conjugation biquandles, which we call $G$-Alexander biquandles and
$G$-Alexander multiple conjugation biquandles, with a relationship with group
cocycles. We give a method to obtain a (biquandle or multiple conjugation
biquandle) cocycle of them from a group cocycle.
