  Let $F_n$ be the free group of rank $n$ with free basis $X=\{x_1,\dots,x_n
\}$. A palindrome is a word in $X^{\pm 1}$ that reads the same backwards as
forwards. The palindromic automorphism group $\Pi A_n$ of $F_n$ consists of
those automorphisms that map each $x_i$ to a palindrome. In this paper, we
investigate linear representations of $\Pi A_n$, and prove that $\Pi A_2$ is
linear. We obtain conjugacy classes of involutions in $\Pi A_2$, and
investigate residual nilpotency of $\Pi A_n$ and some of its subgroups. Let
$IA_n$ be the group of those automorphisms of $F_n$ that act trivially on the
abelianisation, $P I_n$ be the palindromic Torelli group of $F_n$, and $E \Pi
A_n$ be the elementary palindromic automorphism group of $F_n$. We prove that
$PI_n=IA_n \cap E \Pi A_n'$. This result strengthens a recent result of
Fullarton.
