= Solution
Use a <reduced sequence in an HNN extension> for $\gamma$. By repeatedly conjugating a prefix to the other end and reducing any resulting pinch, one obtains a <cyclically reduced sequence in an HNN extension>: either an element of the base group, or a sequence of positive stable-letter length with no pinch even across its cyclic junction. For example, after conjugating away the initial base coefficient, write it as $t^{\epsilon_1}g_1\cdots t^{\epsilon_k}g_k$. A pinch across the junction can occur only if $\epsilon_k=-\epsilon_1$ and $g_k\in C_{\epsilon_k}$. Moving the first stable letter to the end then exposes that pinch and reduces the stable-letter length by two. Iteration must terminate.
If the resulting cyclically reduced sequence has $k>0$, every positive power remains reduced and has stable-letter length equal to that power times $k$. By <Britton's lemma>, none is the identity. Thus a finite-order element must be conjugate into the base group $G$. Since $G$ embeds in the <HNN extension>, the order of a base-group element is unchanged, and conjugation also preserves order. Therefore
$$
\boxed{\gamma\text{ has finite order}\iff\gamma\text{ is conjugate to a finite-order element of }G.}
$$
The argument also applies to several stable letters: cyclic pinches must involve a stable letter and its own inverse.
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