No. We can prove this by induction. Let $G$ be the smallest $2$-group in which this situation occurs. Then $G$ has a normal central subgroup $N$ of order $2$.
If $N$ has trivial intersection with the subgroup $\langle a,t \rangle = D_{16}$, then the same situation occurs in $G/N$, contradicting the minimality of $G$.
So that intersection must be nontrivial, and hence $N \le \langle a,t \rangle$, and then we must have $N = Z(\langle a,t \rangle) = \langle t^4 \rangle$.
But then $t^4 \in Z(G)$, contradicting the assumption that it is conjugate in $G$ to $a$.
The situation you describe can occur in finite groups, such as in simple groups ${\rm PSL}(2,q)$ for some prime powers $q$, such as $q=17$, but not in finite $2$-groups.
