Let C_2 be the group of order 2. We consider a nontrivial action of C_2 on the spectrum Y:=S/2 smash S/\eta. This can also be viewed as the complex points of a finite real-motivic spectrum. We will show that one of the classical v_1-self-maps of Y can be lifted to a C_2 equivariant self-map as well as a real-motivic self-map. Further, we will discuss that the cofiber of the self-map of the R-motivic lift of Y is a realization of the real-motivic Steenrod subalgebra A(1). We will also discuss the question of finding other realizations of A(1).
This is joint work with Prasit Bhattacharya and Bertrand Guillou.