The dihedral group $D_{16}$ of order 16 has a presentation $D_{16}= \langle a,t \ | \ a^2=t^8=atat=e\rangle$.

**Question**: Does there exist a finite 2-group $G$ containing $D_{16}$ as a subgroup, and an element $g \in G$ such that $gag^{-1}=t^4$? Bonus pats-on-the-back if $G$ has order 64.

An obvious reduction: one can assume that $G =\langle D_{16},g\rangle$.

An obvious constraint: $D_{16}$ cannot be normal in $G$ (so $G$ can't have order 32).

[This question has come up in investigations of the Balmer spectrum of $G$-equivariant stable homotopy for finite $p$-groups $G$. Like Dr. Frankenstein, I am looking for interesting subjects to experiment on, and my student Chris Lloyd is serving as the able assistant to the mad scientist.]