Let's extract a clear question (about spectra in general) from your question, and then answer it. Let $E$ be any spectrum.
There is the degree $p$ map $p:S\to S$ from the sphere spectrum to itself. Smashing with $E$, this gives a map $E\to E$; we also call this map $p$. It multiplies elements of $\pi_n(E)$ by $p$. Denote its spectrum cofiber by $E/pE$. The cofibration sequence $E\to E\to E/pE$ is also a fibration sequence, because we are talking about spectra. From the exact sequence $$ \dots \to \pi_n(E)\to \pi_n(E)\to \pi_n(E/pE)\to \pi_{n-1}(E)\to \pi_{n-1}(E)\to \dots $$ you get what I think you want: an exact sequence $$ 0\to coker(p)\to \pi_n(E/pE)\to ker(p)\to 0 $$ where the group on the left is $$ coker (p:\pi_n(E)\to \pi_n(E))=\pi_n\otimes \mathbb F_p $$ (it could also be called $\pi_n(E)/p$) and the group on the right is $$ ker (p:\pi_{n-1}(E)\to \pi_{n-1}(E))=Tor(\pi_{n-1},\mathbb F_p) $$ (I think you are also calling it $\pi_{n-1}(E)[n]$$\pi_{n-1}(E)[p]$.)
$\pi_n(E/pE)$ is called the $n$th mod $p$ homotopy group of $E$.
Curiously, it can have elements of order $4$ if $p=2$.
