Here is a basic theorem from locally compact topological groups. Suppose $G$ is a locally compact abelian topological group with closed subgroup $H$, and Haar measures $dg$ and $dh$. Suppose that $f: G \rightarrow \mathbb C$ is integrable. Then for almost all $g \in G$, the integral $\int\limits_H f(gh)dh$ converges absolutely and becomes an integrable function of $gH$ on the quotient group $G/H$ with respect to its Haar measure $d\bar{g}$. Furthermore, we have
$$\int\limits_G f(g)dg = \int\limits_{G/H} \int\limits_H f(gh)dh d\bar{g}$$
This is proved first when $f$ is continuous and compactly supported, and then extended to arbitrary $f \in L^1(G)$ by density.
I was wondering whether there is anything more about this that can be said when $f$ is not only integrable, but also continuous on $G$. If it makes the results nicer, let's assume $G$ is $\sigma$-compact. If it makes it even nicer, let's assume $G = \mathbb R$ and $H = \mathbb Z$.
Assuming $f: G \rightarrow \mathbb C$ is continuous and integrable...
Is $\int\limits_H f(gh)dh$ absolutely convergent for all $g \in G$?
Is $g \mapsto \int\limits_H f(gh)dh$ continuous?
If $\int\limits_H f(gh)dh$ is absolutely convergent for all $g \in G$, is $g \mapsto \int\limits_H f(gh)dh$ continuous?