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For a finitely generated group $G=⟨x_1,\dots ,x_r⟩$, every quotient group is generated by the images of the $x_i$. Thus finite generation passes to quotients.

For a group extension $1\to N\to G\to Q\to 1$, suppose $N$ and $Q$ are finitely generated. Choose finite group generators of $N$ and lifts to $G$ of finite group generators of $Q$. For any $g\in G$, a word in the lifts has the same image as $g$ in $Q$, so their difference belongs to $N$. Thus the two finite sets together generate $G$, proving preservation under extensions.

For a subgroup $H\le G$ of finite index, choose representatives $T$ of the right cosets $H\backslash G$, with $1$ representing $H$, and a finite symmetric generating set $X$ for $G$. Write $\overline{g}$ for the chosen representative of $Hg$. The finitely many Schreier generatorsbelong to $H$. If $h=x_1\cdots x_k\in H$, set $t_j=\overline{x_1\cdots x_j}$, so $t_0=t_k=1$. ThenEach factor is a listed generator because $H{x}_1\cdots x_j=H{t}_{j-1}{x}_j$. Therefore $H$ is finitely generated. This proves Schreier's lemma directly.

An infinite locally finite group isIts elements are binary sequences of finite support, with coordinatewise addition. It is infinite because the unit vectors are distinct. Any finite collection of elements has its supports contained in one finite set of coordinates, and the subgroup it generates lies in the corresponding finite product of copies of $C_2$. Hence every finitely generated subgroup is finite.

Local finiteness passes to subgroups because a finitely generated subgroup of a subgroup is also one of the ambient group. It passes to quotients as well: lift finitely many group generators of a subgroup in the quotient; the subgroup generated by their lifts is finite, and its image is exactly the desired subgroup.

For local finiteness is closed under extensions, suppose $N$ and $Q=G/N$ are locally finite, and take a finitely generated subgroup $K\le G$. Its image $\overline{K}\le Q$ is finitely generated, hence finite. The map $K\to \overline{K}$ has kernel $K\cap N$, soThe finite-index result already proved makes $K\cap N$ finitely generated. Local finiteness of $N$ makes it finite, and thenSince $K$ was arbitrary, $G$ is locally finite. Thus finite generation is preserved by quotients, extensions and finite-index subgroups; local finiteness is preserved by subgroups, quotients and extensions.

Take a CW complex with one vertex, loops labelled $a,b$, and one two-cell attached along $a^2{b}^3{a}^3{b}^2$. The Seifert-van Kampen theorem gives the required fundamental group. Connected double covering spaces correspond to index-two subgroups, equivalently kernels of surjective homomorphisms $G\to C_2$. The relation imposes $5a+5b=0$ modulo two, so the unique nonzero possibility sends both $a,b$ to one. Thus there is a unique connected double cover up to covering isomorphism.

Construct it with two vertices; both labelled edges switch sheets, and attach the two lifts of the two-cell. Contract one of the $a$ edges to a spanning tree. With coset representatives $1,a$, the remaining Schreier generators can be taken as $A=a^2$, $B=b{a}^{-1}$ and $C=ab$. Reading the two lifts of the attaching word, starting on the two respective sheets, gives $ABCB{A}^{2}BC$ and $ACBCACB$. ThusFor example, a letter $a$ contributes nothing on the first sheet and $A$ on the second; a letter $b$ contributes $B$ on the first and $C$ on the second. Both letters then switch sheets, which directly verifies the two relators.

For a subgroup $H$ of a group generated by a set $S$, choose representatives $t$ of right cosets $Ht$. If $\overline{tg}$ represents $Htg$, then $tg{\overline{tg}}^{-1}$ lies in $H$. These Schreier generators generate $H$: inserting successive coset representatives along a word telescopes the word into such factors. Lifted relators can be rewritten in the same factors.