Summary: We initiate the study of outer automorphism groups of special groups \(G\), in the Haglund-Wise sense. We show that \(\operatorname{Out}(G)\) is infinite if and only if \(G\) splits over a co-abelian subgroup of a centraliser and there exists an infinite-order ‘generalised Dehn twist’. Similarly, the coarse-median preserving subgroup \(\operatorname{Out}_{\mathrm{cmp}}(G)\) is infinite if and only if \(G\) splits over an actual centraliser and there exists an infinite-order coarse-median-preserving generalised Dehn twist. The proof is based on constructing and analysing non-small, stable \(G\)-actions on \(\mathbb{R} \)-trees whose arc-stabilisers are centralisers or closely related subgroups. Interestingly, tripod-stabilisers can be arbitrary centralisers, and thus are large subgroups of \(G\). As a result of independent interest, we determine when generalised Dehn twists associated to splittings of \(G\) preserve the coarse median structure.