  The main focus of this paper is the complete enumeration of self-dual abelian
codes in non-principal ideal group algebras $\mathbb{F}_{2^k}[A\times
\mathbb{Z}_2\times \mathbb{Z}_{2^s}]$ with respect to both the Euclidean and
Hermitian inner products, where $k$ and $s$ are positive integers and $A$ is an
abelian group of odd order. Based on the well-know characterization of
Euclidean and Hermitian self-dual abelian codes, we show that such enumeration
can be obtained in terms of a suitable product of the number of cyclic codes,
the number of Euclidean self-dual cyclic codes, and the number of Hermitian
self-dual cyclic codes of length $2^s$ over some Galois extensions of the ring
$\mathbb{F}_{2^k}+u\mathbb{F}_{2^k}$, where $u^2=0$. Subsequently, general
results on the characterization and enumeration of cyclic codes and self-dual
codes of length $p^s$ over $\mathbb{F}_{p^k}+u\mathbb{F}_{p^k}$ are given.
Combining these results, the complete enumeration of self-dual abelian codes in
$\mathbb{F}_{2^k}[A\times \mathbb{Z}_2\times \mathbb{Z}_{2^s}]$ is therefore
obtained.
