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What is the right definition of equivariant Cech cohomology is so that given a $G$-space $X$, $H^1_G(X;H)$ classifies $G$-equivariant principal $H$-bundles on $X$?

The version I worked up naively requires keeping track of different $G$-space structures on $H$. I'm having two problems with this:

  1. It assumes that given a $G$-equivariant principal H-bundle, I can canonically associate an action of $G$ on $H$ to it. I haven't been able to show this is true.
  2. My definition gets me into trouble when I want to build long exact sequences in cohomology from short exact sequences of coefficient groups.

If there's a good reference on this, or a quick explanation, I'd really appreciate it.
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What is the right notion of equivariant Cech cohomology?

What is the right definition of equivariant Cech cohomology is so that given a $G$-space $X$, $H^1_G(X;H)$ classifies $G$-equivariant principal $H$-bundles on $X$?

The version I worked up naively requires keeping track of different $G$-space structures on $H$. I'm having two problems with this:

  1. It assumes that given a $G$-equivariant principal H-bundle, I can canonically associate an action of $G$ on $H$ to it. I haven't been able to show this is true.
  2. My definition gets me into trouble when I want to build long exact sequences in cohomology from short exact sequences of coefficient groups.

If there's a good reference on this, or a quick explanation, I'd really appreciate it.