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Let $k$ be an algebraically closed field of characteristic $2$, let $G$ be afinite group and let $B$ be the principal $2$-block of $kG$ with a dihedral ora generalised quaternion defect group $P$. Let also $\mathcal{T}(B)$ denote thegroup of splendid Morita auto-equivalences of $B$. We show that \begin{align*}\mathcal{T}(B)\cong \mathrm{Out}_P(A)\rtimes \mathrm{Out}(P,\mathcal{F})\,,\end{align*} where $\mathrm{Out}(P,\mathcal{F})$ is the group of outerautomorphisms of $P$ which stabilize the fusion system $\mathcal{F}$ of $G$ on$P$ and $\mathrm{Out}_P(A)$ is the group of algebra automorphisms of a sourcealgebra $A$ of $B$ fixing $P$ modulo inner automorphisms induced by$(A^P)^\times$.

The group of splendid Morita equivalences of principal $2$-blocks with dihedral and generalised quaternion defect groups