Normalizer of a Cartan subgroup (awaiting review)

For $p>2$ the normalizer of a Cartan subgroup of $\GL_2(\F_p)$ is a maximal subgroup of $\GL_2(\F_p)$ that contains a Cartan subgroup with index 2. It is the normalizer in $\GL_2(\F_p)$ of the Cartan subgroup it contains.

For $p=2$ the Cartan subgroups of $\GL_2(\F_2)$ are already normal and we instead define the normalizer of a Cartan subgroup to be a group that contains a Cartan subgroup with index 2. This means that the normalizer of a split Cartan subgroup of $\GL_2(\F_2)$ has order 2 (which makes it conjugate to the Borel subgroup), while the normalizer of a non-split Cartan subgroup of $\GL_2(\F_2)$ has order 6 (which makes it all of $\GL_2(\F_2)$).