I have a question about a reduction argument from 
Jarden's and Lubotzky's paper  ['Elementary equivalence of
profinite groups'][1] in Lemma 1.1 on page 3: 

*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 $G$.* $\tag{L}$  

 The proof begins with a reduction step I
not understand:
 

*Proof*: 
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 $G/M \cong A$. [..]        $\tag{L'}$

Assume we have proved (L'). Why the claim (L) of the lemma follows
immediately from (L'); in other words why is sufficient to show
only claim (L'), which is seemingly weaker as the claim (L)?

I have already asked identical question in [MathStackEx][2]
but a comment by Noah Schweber stresses another important viewpoint: 
That this lemma 1.1 is an elementary consequence of a much stronger
statement about structures in context of finite languages.
That's true, but that was not my original concern: my original concern
is just about the logic of the proof itself: why (L') *implies* (L).


  [1]: http://132.64.72.10/~alexlub/PAPERS/Elementary%20equivalence%20of%20profinite%20groups/Elementary%20equivalence%20of%20profinite%20groups.pdf
  [2]: https://math.stackexchange.com/questions/3911907/lemma-from-paper-on-elementary-equivalence-of-profinite-groups-by-jarden-lubot