  This paper is the result of a research project completed in the context of
the first author's Undergraduate Student Research Award from the Natural
Sciences and Engineering Research Council of Canada (NSERC).
  We prove a nesting phenomenon for twisted Frobenius extensions. Namely,
suppose $R \subseteq B \subseteq A$ are graded superrings such that $A$ and $B$
are both twisted Frobenius extensions of $R$, $R$ is contained in the center of
$A$, and $A$ is projective over $B$. Our main result is that, under these
assumptions, $A$ is a twisted Frobenius extension of $B$. This generalizes a
result of Pike and the second author, which considered the case where $R$ is a
field.
