This paper studies time regularity for the random walk governed by a probability measure $\mu$ on a locally compact, compactly generated group $G$. If $\mu$ is eventually coset aperiodic on $G$ and satisfies certain additional conditions, we establish that the associated Markov operator $T_{\mu}$
is analytic in $L^2(G)$, that is, one has an estimate
$\|(I-T_{\mu}) T_{\mu}^n \| \leq cn^{-1}$, $n\in \mathbb{N}$,
in $L^2$ operator norm. Alternatively, if $\mu$ is irreducible with period $d$ and satisfies certain conditions, we show that $T_{\mu}^d$ is analytic in $L^2(G)$.
To obtain these results, we develop a number of interesting algebraic and spectral properties of coset aperiodic or irreducible measures on groups.