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The algebraic Joker module was originally described in the 1970s by Adams andPriddy and is a $5$-dimensional module over the subHopf algebra$\mathcal{A}(1)$ of the mod $2$ Steenrod algebra. It is a self-dual endotrivialmodule, i.e., an invertible object in the stable module category of$\mathcal{A}(1)$. Recently it has been shown that no analogues exist for$\mathcal{A}(n)$ with $n>1$. Using iterated doubling this also gives aniterated double which is an $\mathcal{A}(n)$-module but not stably invertible.In previous work the author showed that for $n=1,2,3$ these iterated doubleswere realisable as cohomology of CW spectra, but no such realisation existedfor $n>3$.  The main point of the paper is to show that in the height $2$ chromaticcontext, the Morava $K$-theory of double Jokers realise an exceptionalendotrivial module over the quaternion group of order $8$ that only exists overa field of characteristic $2$ containing a primitive cube root of unity. Thishas connections with certain Massey products in the cohomology of thequaternion group.

Endotrivial modules for the quaternion group and iterated Jokers in chromatic homotopy theory