  Dolfi, Guralnick, Praeger and Spiga asked if there exist infinitely many
primitive groups of twisted wreath type with nontrivial coprime subdegrees.
Here we settle this question in the affirmative. We construct infinite families
of primitive twisted wreath permutation groups with nontrivial coprime
subdegrees. In particular, we define a primitive twisted wreath group $G(m,q)$
constructed from the nonabelian simple group $\text{PSL}(2,q)$ and a primitive
permutation group of diagonal type with socle $\text{PSL}(2,q)^m$, and
determine all values of $m$ and $q$ for which $G(m,q)$ has nontrivial coprime
subdegrees. In the case where $m=2$ and $q\notin\{7,11,29\}$ we obtain a full
classification of all pairs of nontrivial coprime subdegrees.
