= Solution
For a <finitely generated group> $G=\langle x_1,\ldots,x_r\rangle$, every <quotient group> is generated by the images of the $x_i$. Thus finite generation passes to quotients.
For a <group extension> $1\to N\to G\to Q\to1$, 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> $H\leq G$ 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 generators>
$$
t x\,\overline{tx}^{-1}\qquad(t\in T,\ x\in X)
$$
belong to $H$. If $h=x_1\cdots x_k\in H$, set $t_j=\overline{x_1\cdots x_j}$, so $t_0=t_k=1$. Then
$$
h=(t_0x_1t_1^{-1})(t_1x_2t_2^{-1})\cdots(t_{k-1}x_kt_k^{-1}).
$$
Each factor is a listed generator because $Hx_1\cdots x_j=Ht_{j-1}x_j$. Therefore $H$ is finitely generated. This proves <Schreier's lemma> directly.
An infinite <locally finite group> is
$$
L=\bigoplus_{j=1}^{\infty}C_2.
$$
Its 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> $K\leq G$. Its image $\overline K\leq Q$ is finitely generated, hence finite. The map $K\to\overline K$ has kernel $K\cap N$, so
$$
[K:K\cap N]=|\overline K|<\infty.
$$
The finite-index result already proved makes $K\cap N$ finitely generated. Local finiteness of $N$ makes it finite, and then
$$
|K|=|K\cap N|\,|\overline K|<\infty.
$$
Since $K$ was arbitrary, $G$ is locally finite. Thus \b[finite generation is preserved by quotients, extensions and finite-index <subgroups>; local finiteness is preserved by <subgroups>, quotients and extensions].
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