I am trying to compute $ [\mathbb{HZ}/4,\mathbb{HZ}/4 ]$ the mod 4 Steenrod Algebra. For some reason, I need to work it out till dimension 6 or so. My approach is to use the cofiber sequence

$\mathbb{HZ}/4 \to \mathbb{HZ}/2 \xrightarrow{Sq^{1}}\Sigma \mathbb{HZ}/2 $

twice.

I did the computation till first five degrees. A description would be as follows, Let $g$ be the generator of $\mathbb{Z}/4$ in $[\mathbb{HZ}/4,\mathbb{HZ}/4]_{0}$, for convenience let us denote $g' = 2g$. So whenever there is an element with $g'$ would mean that it has $2$ torsion.

Degree | elements

$ 1: \beta g$

$ 2: Sq^{2} g' $ (this would mean that $2Sq^{2} g' =0$)

$ 3 : Sq^{3}g$, (but satisfies $2Sq^{3}g =0$ ), $Sq^{2} \beta g'$

$ 4 : Sq^{3} \beta g$( $2 Sq^{3} \beta g = 0$), $Sq^{4}g'$

$ 5 : Sq^{5}g$( $2 Sq^{5} g = 0$), $Sq^{4} \beta g' = 0$

I do not know if they are write, but if somebody has done it please verify. I suspect the all the groups $\mathbb{Z}/4$ in $[\mathbb{HZ}/4,\mathbb{HZ}/4]_{n}$ are two torsion except $n=0,1$. Is this some sort of known result? Also another thing that I am worrying about is the equivalent Cartan formula in Mod 4 Steenrod algebra? Is there a way to detect the Cartan formula?