TITLE: A finite p-group question: can this happen?
QUESTION [6 upvotes]: 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 I posed, a minimal counterexample would have $G = \langle H,g\rangle$, and $H$ not normal in $G$.

REPLY [12 votes]: 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 $r(K_2,H)=1$.