Local powering-invariance is not quotient-transitive in solvable group

Statement
It is possible to have a group $$G$$ and subgroups $$H,K$$ of $$G$$ such that $$H$$ is a local powering-invariant normal subgroup of $$G$$ and $$K/H$$ is a local powering-invariant subgroup in the quotient group $$G/H$$, but $$K$$ is not a local powering-invariant subgroup of $$G$$.

Similar facts

 * Second center not is local powering-invariant in solvable group
 * Center not is quotient-local powering-invariant in solvable group

Opposite facts

 * Local powering-invariance is quotient-transitive in nilpotent group
 * Quotient-local powering-invariance is quotient-transitive
 * Local powering-invariant over quotient-local powering-invariant implies local powering-invariant

Facts used

 * 1) uses::Second center not is local powering-invariant in solvable group
 * 2) uses::Center is local powering-invariant

Proof
Facts (1) and (2) together give us the result. Namely, any example for Fact (1) automaticlaly gives an example of Fact (2) by setting $$H$$ as the center and $$K$$ as the second center.