Normality is not transitive for any nontrivially satisfied extension-closed group property
From Groupprops
Statement
Suppose p is a group property such that there is a nontrivial group satisfying pband p is closed under taking extensions (in other words, if a normal subgroup and its quotient group both satisfy p, so does the whole group). Then, there exists a group G satisfying p, a normal subgroup K of G satisfying p, and a normal subgroup H of K, also satisfying p, that is not normal in G.
In fact, for any nontrivial group H satisfying p, we can find K and G so that the above occurs.
