One of the open questions that has emerged in the study of the projective Schur group PS (F) of a field F is whether or not PS (F) is an algebraic relative Brauer group over F, i.e. does there exist an algebraic extension L / F such that PS (F) = Br (L / F)? We show that the same question for the Schur group of a number field has a negative answer. For the projective Schur group, no counterexample is known. In this paper we prove that PS (F) is an algebraic relative Brauer group for all Henselian valued fields F of equal characteristic whose residue field is a local or global field. For this, we first show how PS (F) is determined by PS (k) for an equicharacteristic Henselian field with arbitrary residue field k.