For a centre-by-metabelian pro-$p$ group $G$ of type $\text{FP}_{2m}$, for some $m\geqslant 1$, we show that $\sup _{M\in {\mathcal{A}}}$ rk $H_{i}(M,\mathbb{Z}_{p})<\infty$, for all $0\leqslant i\leqslant m$, where ${\mathcal{A}}$ is the set of all subgroups of $p$-power index in $G$ and, for a finitely generated abelian pro-$p$ group $V$, rk $V=\dim V\otimes _{\mathbb{Z}_{p}}\mathbb{Q}_{p}$.

For a finite abelian $p$-group $A$ of rank $d\,=\,\dim\,A/pA$, let ${{\mathbb{M}}_{A}}\,:=\,\text{lo}{{\text{g}}_{p}}\,{{\left| A \right|}^{1/d}}$ be its (logarithmic) mean exponent. We study the behavior of the mean exponent of $p$-class groups in pro-$p$ towers $\text{L/K}$ of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-$p$ towers in which the mean exponent of $p$-class groups remains bounded. Several explicit examples are given with $p\,=\,2$. Turning to group theory, we introduce an invariant $\underline{\mathbb{M}}\left( G \right)$ attached to a finitely generated pro-$p$ group $G$; when $G\,=\,\text{Gal}\left( \text{L/K} \right)$, where $L$ is the Hilbert $p$-class field tower of a number field $K$, $\underline{\mathbb{M}}\left( G \right)$ measures the asymptotic behavior of the mean exponent of $p$-class groups inside $\text{L/K}$. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.