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 $G$.



A finite p-group question: can this happen?
It does not follow. Let $G=Z\_4\wr Z\_2$, or equivalently $$G=\langle a,b,t|a^4=b^4=[a,b]=t^2=1, a^t=b\rangle.$$ Let $H=\langle a,b^2\rangle$, $K\_1=\langle a^2\rangle$, and $K\_2=\langle b^2\rangle$. Then $K\_1^t=K\_2$, $r(K\_1,H)=2$, and $r(K\_2,H)=1$. 


