  Let $\mathbb{F}G$ denote the group algebra of the group $G$ over the field
$\mathbb{F}$ with $char(\mathbb{F})\neq 2$. Given both a homomorphism
$\sigma:G\rightarrow \{\pm1\}$ and a group involution $\ast: G\rightarrow G$,
an oriented involution of $\mathbb{F}G$ is defined by $\alpha=\Sigma\alpha_{g}g
\mapsto \alpha^\circledast=\Sigma\alpha_{g}\sigma(g)g^{\ast}$. In this paper,
we determine the conditions under which the group algebra $\mathbb{F}G$ is
normal, that is, conditions under which $\mathbb{F}G$ satisfies the
$\circledast$-identity $\alpha\alpha^\circledast=\alpha^\circledast\alpha$. We
prove that $\mathbb{F}G$ is normal if and only if the set of symmetric elements
under $\circledast$ is commutative.
