  Split group codes are a class of group algebra codes over an abelian group.
They were introduced in 2000 by Ding, Kohel and Ling as a generalization of the
cyclic duadic codes. For a prime power q and an abelian group G of order n such
that n and q are coprime, consider the group algebra F_{q^2}[G^{*}] of F_{q^2}
over the dual group G^{*} of G. We prove that every ideal code in
F_{q^{2}}[G^{*}] whose extended code is Hermitian self-dual is a split group
code. We characterize the orders of finite abelian groups G for which an ideal
code of F_{q^2}[G^{*}] whose extension is Hermitian self-dual exists and derive
asymptotic estimates for the number of non-isomorphic abelian groups with this
property.
