Power sums in $(\mathbb Z/p^a\mathbb Z)^\times$

Let $p$ be an odd prime and $a\geq 1$. Consider the reduced residue system modulo $p^a$, namely the set of integers $x$ such that $$ 1\leq x\leq p^a,\qquad \gcd(x,p)=1 $$

I would like to determine the following power sum modulo $p^a$:

$$ S_k=\sum_{\substack{1\leq x\leq p^a\\ \gcd(x,p)=1}}x^k \pmod{p^a} $$

where $k$ is a positive integer.

Equivalently, this asks for the value of

$$ \sum_{u\in(\mathbb{Z}/p^a\mathbb{Z})^\times}u^k $$

in the ring $\mathbb{Z}/p^a\mathbb{Z}$.

What is the general formula for $S_k$? Is there a short proof using the structure of the group $(\mathbb{Z}/p^a\mathbb{Z})^\times$?