Fuzzy subgroups of the direct product of a generalized quaternion group and a cyclic group of any odd order

Bentea and T\u{a}rn\u{a}uceanu~(An. \c{S}tiin\c{t}. Univ. Al. I. Cuza Ia\c{s}, Ser. Nou\v{a}, Mat., {\bf 54(1)} (2008), 209-220) proposed the following problem: Find an explicit formula for the number of fuzzy subgroups of a finite hamiltonian group of type $Q_8\times \mathbb{Z}_n$ where $Q_8$ is the quaternion group of order $8$ and $n$ is an arbitrary odd integer. In this paper we consider more general group: the direct product of a generalized quaternion group of any even order and a cyclic group of any odd order. For this group we give an explicit formula for the number of fuzzy subgroups.