In Jarden's and Lubotzky's paper 'Elementary equivalence of profinite groups' Lemma 1.1 on page 3 the proof starts with a reduction step I not understand at all. The lemma is:
Lemma 1.1: For each positive integer $n$ and each finite group $A$ of order at most $n$ there exists a sentence $ \theta $ of $\mathcal{L} \text{(group)} $ such that for every group $G$ of order at most $n$ the sentence $ \theta $ holds in $G$ if and only if $A$ is a quotient of $G$. $\tag{L}$
Proof: The proof is based on verification of a seemingly weaker claim:
It suffices to prove that for every positive integer $n$ and for every group $A$ of order $d$ dividing $n$ there exists a sentence $ \theta $ of $\mathcal{L} \text{(group)} $ with the following property: for every group $G$ of order $n$ the sentence $ \theta $ holds in $G$ if and only if $G$ has a normal subgroup $M$ such that $G/M \cong A$. [..] $\tag{L'}$
Why is it sufficient to prove the claim (L') which is seemingly weaker as the claim of the lemma (L)? The claim of the lemma immediately imply the second claim, the reverse implication is not clear.