A subset grouping problem
The problem is NP-complete already for $k=3$ (and I guess also for $k=2$).
We reduce the following problem to it:
For an input $r$-regular graph $G$, decide whether $G$ has an equitable $3$-coloring, i.e., one where the size of each color class is the same.
(I couldn't find a reference for this, but one can easily reduce the classical $3$-coloring problem to this by adding a few extra vertices and edges.)

The reduction is as follows.
For each vertex $v$ of the input graph $G$, define a set $S_v=\{\{u,v\}\mid uv\notin E(G)\}$.
For any subset $X$ of the vertices we have

$$|\cup_{\{v\in X\}} S_v|=|X|(n-1-r)-|X_2|$$

where $X_2$ is the set of non-adjacent pairs in $X$.
If $|X|\ge n/3$, this is at least

$$\frac{n(n-1-r)}3-\binom{\frac n3}{2}$$

with equality only if $|X|=n/3$ and all pairs in $X$ are non-adjacent.
Since in any grouping of the $S_v$ into three groups we have such an $X$, we are done.