Projective Tetrahedral Representation I can embed $A_4$ as a subgroup into $PSL_2(\mathbb{F}_{13})$ (in two different ways in fact). I also have a reduction mod 13 map $$PGL_2(\mathbb{Z}_{13}) \to PGL_2(\mathbb{F}_{13}).$$ My question is:

Is there a subgroup of $PGL_2(\mathbb{Z}_{13})$ which maps to my copy of $A_4$ under the above reduction map?

(I know that one may embed $A_4$ into $PGL_2(\mathbb{C})$, but I don't know about replacing $\mathbb{C}$ with $\mathbb{Z}_{13}$). 

Yes.  Explicitly one has:
$
\newcommand{\ze}{\zeta_3}
\newcommand{\zi}{\ze^{-1}}
\newcommand{\vp}{\vphantom{\zi}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\GL}{\operatorname{GL}}
$
$$ \SL(2,3) \cong
G_1 = \left\langle
\begin{bmatrix} 0   & 1 \vp \\ -1 & 0 \vp \end{bmatrix},
\begin{bmatrix} \ze & 0     \\ -1 & \zi   \end{bmatrix}
\right\rangle
\cong
G_2 = \left\langle
\begin{bmatrix} 0 & 1 \vp \\ -1 & 0 \vp \end{bmatrix},
\begin{bmatrix} 0 & -\zi  \\  1 & -\ze  \end{bmatrix}
\right\rangle
$$
and
$$G_1 \cap Z(\GL(2,R)) = G_2 \cap Z(\GL(2,R)) = Z =  \left\langle\begin{bmatrix}-1&0\\0&-1\end{bmatrix}\right\rangle \cong C_2$$
and
$$G_1/Z \cong G_2/Z \cong A_4$$
This holds over any ring R which contains a primitive 3rd root of unity, in particular, in the 13-adics, $\mathbb{Z}_{13}$.  The first representation has rational (Brauer) character and Schur index 2 over $\mathbb{Q}$ (but Schur index 1 over the 13-adics $\mathbb{Q}_{13}$), and the second representation is the unique (up to automorphism of $A_4$) 2-dimensional projective representation of $A_4$ with irrational (Brauer) character.
You can verify that if $G_i = \langle a,b\rangle$, then $a^2 = [a,a^b] = -1$, $ a^{(b^2)} = aa^b$, and $b^3 = 1$.  Modulo $-1$, one gets the defining relations for $A_4$ on $a=(1,2)(3,4)$ and $b=(1,2,3)$.
