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 $\pi_{n-1}(E)[n]$.)

$\pi_n(E/pE)$ is called the $n$th mod $p$ homotopy group of $E$. 

Curiously, it can have elements of order $4$ if $p=2$.