Random walks on finite groups converging after finite number of steps

Let $P$ be a probability on a finite group $G$, $P^{(n)}=P\ast\ldots\ast P$ ($n$ times) be an $n$-fold convolution of $P$. If $n\rightarrow\infty$, then under mild conditions $P^{(n)}$ converges to the uniform probability $U(g)=\frac 1{|G|}$ $(g\in G)$. We study the case when the sequence $P^{(n)}$ reaches its limit $U$ after finite number of steps: $P^{(k)}=P^{(k+1)}=\dots=U$ for some $k$. Let $\Omega(G)$ be a set of the probabilities satisfying to that condition. Obviously, $U\in\Omega(G)$. We prove that $\Omega(G)\neq U$ for “almost all” non-Abelian groups and describe the groups for which $\Omega(G)=U$. If $P\in \Omega(G)$, then $P^{(b)}=U$, where $b$ is the maximal degree of irreducible complex representations of the group $G$.