If $H = Z(G),$ then $Z(G/H) = 1.$

Question

I know that this statement is false: If $H = Z(G),$ then $Z(G/H) = 1.$

And I have known from a previous question of mine that any non-abelian nilpotent group is a counterexample.

My questions are:

1- Why any non-abelian nilpotent group is a counterexample? what is the idea that we are using to create a counterexample?

2- Could anyone show me the details that show that the quaternions group is a counterexample, please? why is the center of the quaternions $\mathbb Z_{2}$?

Could anyone help me with that, please?

Answer 1

Quotients of nilpotent groups are also nilpotent, and nontrivial nilpotent groups have a nontrivial center practically by definition. For the second question, just use the fact that any group of order $4$ is abelian.