Let all groups here be finite $p$--groups. 

Given $K<H$, let $r(K,H)$ be the smallest $r$ such that there exists a chain of subgroups $K=L_0 \lhd L_1 \lhd \cdots \lhd L_r = H$, such that each $L_i/L_{i-1}$ is cyclic.

**Question**: Let $H$ be a subgroup of $G$.  If $K_1$ and $K_2$ are subgroups of $H$ that are conjugate in $G$, does it follow that $r(K_1,H) = r(K_2,H)$?

As in the last [question](https://mathoverflow.net/questions/334948/a-finite-2-group-containing-the-dihedral-group-of-order-16) I posed, a minimal counterexample would have $G = \langle H,g\rangle$, and $H$ not normal in $G$.