  We characterize Frobenius and separable monoidal algebra extensions $i: R\ra
S$ in terms given by $R$ and $S$. For instance, under some conditions, we show
that the extension is Frobenius, respectively separable, if and only if $S$ is
a Frobenius, respectively separable, algebra in the category of bimodules over
$R$. In the case when $R$ is separable we show that the extension is separable
if and only if $S$ is a separable algebra. Similarly, in the case when $R$ is
Frobenius and separable in a sovereign monoidal category we show that the
extension is Frobenius if and only if $S$ is a Frobenius algebra and the
restriction at $R$ of its Nakayama automorphism is equal to the Nakayama
automorphism of $R$. As applications, we obtain several characterizations for
an algebra extension associated to a wreath to be Frobenius, respectively
separable.
