When is a semilocal group algebra continuous?

It is shown that (i) a semilocal group algebra KG of an infinite nilpotent group G over a field K of characteristic p>0 is CS (equivalently continuous) if and only if G=P×H, where P is a locally finite, infinite p-group and H is a finite abelian group whose order is not divisible by p, (ii) if K is a field of characteristic p>0 and G=P×H where P is an infinite locally finite p-group (not necessarily nilpotent) and H is a finite group whose order is not divisible by p then KG is CS if and only if H is abelian. Furthermore, commutative semilocal group algebra is always continuous and for PI group algebras this holds for local group algebras; however this result is not true, in general.