IfC,Vare Grothendieck categories andF:C→D,G:D→Care functors satisfying convenient conditions, the extension an restriction of torsion theories preserves the compatibility.Moreover, two idempotent kernel functors are compatible inDif, and only if, their extensions are compatible inC.Finally, whenC,Dare symmetric closed, we obtain relations between exact sequences of Picard and Brauer groups for the extension and restriction of torsion theories.