Let $G=(V,E)$ be a finite simple graph. For any positive integer $k \geq 1$, a distance-2 $K_k$-factor is a collection of cliques $K_k$ of $G$ such that for every two vertices $u$ and $v$ in distinct members of the collection, the distance between $u$ and $v$ is at least two.
For $k = 1$, this is known as a distance-2 independent set. For $k = 2$, this is known as an induced matching (also known as a strong matching). These are known concepts that have definitely been studied before.
But has the problem been studied for larger $k$, particularly $k=3$, before? Does it have a more familiar name?