  Let $X$ be a normal variety over a perfect field of positive characteristic
and $B$ a reduced divisor on $X$. We prove that if the Cartier isomorphism on
the log smooth locus of $(X,B)$ extends to the entire $X$, then $(X,B)$
satisfies the logarithmic extension theorem for differential forms. As an
application, we show that the logarithmic extension theorem holds for good
quotients of smooth varieties by actions of reduced linearly reductive group
schemes. In addition, the logarithmic extension theorem for one-forms holds for
singularities of higher codimension under assumptions about Serre's condition.
We also prove that tame quotients satisfy the regular extension theorem.
