Let $M$ be Riemannian manifold and $\tilde M$ be its universal cover (with induced metric). What is the upper bound for $k=\mathop{diam}\tilde M/\mathop{diam} M$ in terms of $m=|\pi_1(M)|$ (or $\pi_1(M)$)?

**Comments:**

There is a similar answered question here, but there cover is NOT universal. So we get $k\leqslant m$, but $k\ll m$ is expected.

The question is open even in case $\pi_1=\mathbb Z_m$ (even asymptotics is not known).

Clearly, $\sup k$ for given finite group $\Gamma=\pi_1(M)$ is an invariant of $\Gamma$. Is it an interesting invariant?

**Examples:**

For $\pi_1=\mathbb Z_{3\cdot 2^n}$ one can make $k\sim n$ or $k=O(\log m)$ (see my answer below).

For $\pi_1=S_n$, one can make $k$ of order $n^2$ or (see Greg's answer). It is much more than $\log m$, but still $k=o(\log^2 m)$.