Generalized symmetries and the dimensional reduction of 6d so SCFTs

We consider the dimensional reduction on a torus of the family of 6d $(1,0)$ SCFTs UV completing an $so(N)$ gauge theory with $N-8$ vector hypermultiplets. These SCFTs are known to possess a rich structure of discrete symmetries, notably 0-form and 1-form symmetries, which often merge to form a higher group structure, both split and non-split. We investigate what happens to this symmetry structure once the theory is reduced on a circle to 5d and on a torus to 4d, especially when a non-trivial Stiefel-Whitney class for the flavor symmetry is turned on. Unlike in Lagrangian theories, here the 1-form symmetries of the 6d theory reduce to non-trivially acting 1-form and 0-form symmetries, and the original higher group structure leads to an extension of the 0-form symmetries.