As a nontrivial extension of the dihedral quandle \(R_3\) of order 3, a Galkin quandle G(A, c) is defined for each pointed abelian group (A, c). In this paper, we construct an extension \(G_Q(A, \gamma )\) of an Alexander quandle Q for each pair \((A, \gamma )\) of an abelian group A and an element \(\gamma \) of a power set \(A^B\) for some quotient set B of Q as a generalization of Galkin quandles; when \(Q = R_3\) , \(G_Q(A, \gamma )\) is isomorphic to a Galkin quandle. We also give a topological interpretation to their colorings, that is, we see that a lift of a Q-coloring \(\rho \) to \(G_Q(A, \gamma )\) has an information of the homology of the (irregular) branched covering space associated with \(\rho \) . Furthermore, we can calculate linking numbers in the covering space as quandle 2-cocycle invariants. As an application, we show that any coloring on the dihedral quandle of order \(2m + 1\) lifts to the \((2m + 1)\) -dimensional hyper-octahedral quandle. In terms of groups, this implies that any surjective representation of a knot group to the dihedral group \(D_{2m+1}\) of order \(2(2m+1)\) admits nontrivial lifts to the \((2m+1)\) -dimensional hyper-octahedral group.
