  Let $F,E\subseteq \mathbb{R}^2$ be two self similar sets. First, assuming $F$
is generated by an IFS $\Phi$ with strong separation, we characterize the
affine maps $g:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $g(F)\subseteq
F$. Our analysis depends on the cardinality of the group $G_\Phi$ generated by
the orthogonal parts of the similarities in $\Phi$. When $|G_\Phi|=\infty$ we
show that any such self embedding must be a similarity, and so (by the results
of Elekes, Keleti and M\'ath\'{e}) some power of its orthogonal part lies in
$G_\Phi$. When $|G_\Phi| < \infty$ and $\Phi$ has a uniform contraction
$\lambda$, we show that the linear part of any such embedding is
diagonalizable, and the norm of each of its eigenvalues is a rational power of
$\lambda$.
  We also study the existence and properties of affine maps $g$ such that
$g(F)\subseteq E$, where $E$ is generated by an IFS $\Psi$. In this direction,
we provide more evidence for a Conjecture of Feng, Huang and Rao, that such an
embedding exists only if the contraction ratios of the maps in $\Phi$ are
algebraically dependent on the contraction ratios of the maps in $\Psi$.
Furthermore, we show that, under some conditions, if $|G_\Phi|=\infty$ then
$|G_\Psi|=\infty$ and if $|G_\Phi|<\infty$ then $|G_\Psi|<\infty$.
