  A ring $R$ is called clean if every element of $R$ is the sum of a unit and
an idempotent. Motivated by a question proposed by Lam on the cleanness of von
Neumann Algebras, Va\v{s} introduced a more natural concept of cleanness for
$*$-rings, called the $*$-cleanness. More precisely, a $*$-ring $R$ is called a
$*$-clean ring if every element of $R$ is the sum of a unit and a projection
($*$-invariant idempotent). Let $\mathbb F$ be a finite field and $G$ a finite
abelian group. In this paper, we introduce two classes of involutions on group
rings of the form $\mathbb FG$ and characterize the $*$-cleanness of these
group rings in each case. When $*$ is taken as the classical involution, we
also characterize the $*$-cleanness of $\mathbb F_qG$ in terms of LCD abelian
codes and self-orthogonal abelian codes in $\mathbb F_qG$.
