  Fix an equilateral triangle group $T_i=\langle a, b; a^i, b^i, (ab)^i\rangle$
with $i\geq6$ arbitrary. Our main result is: for every presentation
$\mathcal{P}$ of every countable group $Q$ there exists an HNN-extension
$T_{\mathcal{P}}$ of $T_i$ such that $\operatorname{Out}(T_{\mathcal{P}})\cong
Q$. We construct the HNN-extensions explicitly, and examples are given. The
class of groups constructed have nice categorical and residual properties. In
order to prove our main result we give a method for recognising malnormal
subgroups of small cancellation groups, and we introduce the concept of
"malcharacteristic" subgroups.
