I am currently reading Shalika's article "Representation of the two by two unimodular group over local fields" and various other related articles, which deal with the classification of complex representation of reductive groups over local rings. It is cumbersome, that some authors consider only local rings with residue fields of characteric $p \neq 2$.


Why is $p=2$ special here?


Perhaps some words about the strategy, which one should use - at least from my perspective:
Consider a finite extension $K$ of $\mathbb{Q}\_p$. Let $\mathfrak{o}$ be the ring of integers in $K$ and $\mathscr{p}$ its maximal ideal. We want to classify all representation of $\mathrm{GL}\_2( \mathfrak{o})$. Since we deal with a pro-$p$-group, the representations live on $\mathrm{GL}\_2( \mathfrak{o}/\mathfrak{p}^n)$ for some $n>0$. We proceed by induction over $n$:


1) Classify all representation of $\mathrm{GL}\_2(\mathbb{F}\_q)$, where $\mathbb{F}\_q$ is the residue field.


2) Use Mackey's formalism for the group extension (non split)
$$ 0 \rightarrow M\_{2\times2}(\mathbb{F}\_q) \rightarrow \mathrm{GL}\_2( \mathfrak{o}/\mathfrak{p}^n) \rightarrow \mathrm{GL}\_2( \mathfrak{o}/\mathfrak{p}^{n-1}) \rightarrow 0.$$


Apparently the difficulties do already arise in step 1, since Piatesko-Shapiro in his lecture "Complex representations of $\mathrm{GL}\_2$ over a finite field" only considers characteristic $\neq 2$.



Why is p=2 special, if we want to classify cplx. representation of GL2(Zp)?
Shalika's paper that you mention deals not with $\mathrm{GL}\_2(\mathfrak{o})$, but with $\mathrm{SL}\_2(\mathfrak{o})$. The representation theory of $\mathrm{GL}\_2(\mathfrak{o})$ is uniform in the residue characteristic $p$, in the sense that for any $r,n\in\mathbb{N}$, there exist polynomials $f\_{r,n}(X)\in\mathbb{Z}[X]$ such that the number of irreducible representations of $\mathrm{GL}\_2(\mathfrak{o}/\mathfrak{p}^r)$ of dimension $n$, is given by $f\_{r,n}(q)$, where $q$ is the residue cardinality of $\mathfrak{o}$ (see U. Onn: *Representations of automorphism groups of finite $\mathfrak{o}$-modules of rank two*). Moreover, Onn gives a construction of the representations of $\mathrm{GL}\_2(\mathfrak{o})$ which works uniformly for any residue characteristic $p$.


In contrast, the representation theory of $\mathrm{SL}\_2(\mathfrak{o})$ depends dramatically on the parity of the residue characteristic $p$. The case when $p$ is odd has been known for a long time, but the case when $p=2$ has so far only been done when $\mathfrak{o}=\mathbb{Z}\_2$ by Nobs and Wohlfart, and is very different. There are several reasons for this difference, ranging from the trivial to the less so. Some of these are:


1. Every element in $\mathbb{F}\_{2^m}$ is a square, while this is not the case for $\mathbb{F}\_{q}$ when $q$ is odd.
2. Two by two scalar matrices over $\mathbb{F}\_{2^m}$ have trace zero,
3. The presence of wild ramification,
4. The fact that Hensel's lemma cannot be applied in the same way as when $p$ is odd to obtain the number of solutions in $\mathfrak{o}/\mathfrak{p}^r$ of equations of the form $x^2+sxy+\Delta y^2 = 1$, where $s,\Delta\in \mathfrak{p}$.


Of course, wild ramification is present also for $\mathrm{GL}\_2(\mathfrak{o})$, so one cannot say that this plays a role in itself, but rather it seems that it is the combination of the above factors (and possibly others), which accounts for the dependence on the parity mentioned above. 


