Let G be a finite group. Let $\theta : G \to D_8$ be an onto homomorphism. Assume that $|\ker(\theta)| = 4$. Let $H = ⟨a^{2}⟩ \leq D_8$ and let $L = θ^{−1}(H)$, where $D_8$ is the dihedral group of order $8, D_8 =⟨{a,b:a^{4}=e=b^{2},bab=a^{3}}⟩$
What can you say about $|G|$?
What can you say about $|L|$? Can you find a familiar group that is isomorphic to $L/\ker(θ)$?
Is $L \lhd G$? If it is, find a familiar group that is isomorphic to $G/L$.
The first part I said the following. By the First homomorphism theorem, we have $G/\ker(\theta) \cong \operatorname{Im}(\theta)$ = $D_8$ as $\theta$ is onto. By Lagrange’s theorem, $|G|/|\ker(\theta)| = |D_8|$ and so $|G|=|\ker(θ)| \cdot |D_8| = 4 \cdot 8=32$·
The second part, By the Correspondence theorem, $\ker(\theta) \leq L$ and $L/\ker(\theta) \cong H$. As $H = ⟨(a^{2})⟩, H$ is a cyclic group of order $2$ and so $H \cong \Bbb Z_2$. Thus $|L| = |\ker(\theta)|\cdot |H| = 4 \cdot 2 = 8$ and $L/\ker(\theta) \cong \Bbb Z_2$
Finally for the last question I have the following. Since $|D_8 : H| = 4$, we have $H \lhd D_8$. As $\theta$ is onto, by the Correspondence theorem again, $L \lhd G$ and by the Third homomorphism theorem, $G/L \cong D_8/H \cong \Bbb Z_4$.
I think most of this is right I just want clarification.
It all seems correct to me. Well done.
As is customary for a proof-verification question, an answer that is more than just a yes/no response is more polite & helpful.
So just let me mention that the Correspondence Theorem is sometimes referred to as the "Fourth Isomorphism Theorem" for reasons such as a series of questions such as yours.