$\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\F}{\mathbb{F}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Rep}{\operatorname{Rep}}$
First, recall the classical Kummer congruences for Bernoulli numbers: If $h \equiv k \not\equiv 0 \bmod{p^n(p-1)}$ then \begin{equation} (1-p^{h-1}) \frac{B_h}{h} \equiv (1-p^{k-1})\frac{B_k}{k} \bmod{p^{n+1}}. \end{equation}
This establishes a sort of $p$-adic continuity which is what Kubota and Leopoldt use to construct their first $p$-adic $L$-function.
I was thinking about $\F_p$-representations of the finite group $G = \Z/p\Z \rtimes (\Z/p\Z)^\times$ where the action of the right on the left is by multiplication. Let $V$ be the $2$-dimensional $\F_p$-representation of $G$ defined by \begin{equation} (b, u) \mapsto \begin{pmatrix}u & b \\ 0 & 1\end{pmatrix} \in \text{GL}_2(\F_p). \end{equation}
I noticed that in the $\F_p$-representation ring of $G$, $\Rep(G, \F_p)$, the symmetric powers of $V$ satisfy some type of "$p$-adic continuity": If $h \equiv k \bmod{p^n(p-1)}$ then \begin{equation} \Sym^h V \equiv \Sym^k V \bmod{p^{n-1}\F_p[G]}. \end{equation}
Is there a way to "$p$-adically interpolate" these symmetric powers? Can we give any meaning to $\Sym^aV$ for a $p$-adic integer $a$?