= Solution
\b[False, even when the HNN extension also has a finite presentation.] Take the base $P=F_m\times F_m$ and its <Mihailova subgroup> $L$ from Question 4(b). Its <word problem for a group> is soluble: project a word to each free factor and freely reduce both projections, as in part (iv). The base is finitely presented, but the <HNN extension>
$$
G=\langle P,t\mid t^{-1}\ell t=\ell\ (\ell\in L)\rangle
$$
has insoluble word problem by the explicit reduction in Question 4(b). Namely, for an input word $w$ in the presentation with insoluble word problem, set $p=(1,w)$; then
$$
\boxed{t^{-1}ptp^{-1}=1\text{ in }G\quad\Longleftrightarrow\quad p\in L
\quad\Longleftrightarrow\quad w=1\text{ in }H.}
$$
The obstruction is undecidable membership in the associated subgroup, not undecidable equality in the base. Soluble base word problem alone does not let one recognize the pinches needed for a <normal form theorem for an HNN extension> algorithm.
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