  In this paper, we establish sharp two-sided heat kernel estimates for a large
class of symmetric Markov processes in exterior $C^{1,\eta}$ open sets for all
$t> 0$. The processes are symmetric pure jump Markov processes with jumping
kernel intensity $$\kappa(x, y)\psi(|x-y|)^{-1}|x-y|^{-d-\alpha}$$ where
$\alpha\in(0,2)$, $\psi$ is an increasing function on $[ 0, \infty)$ with
$\psi(r)=1$ on $0<r\le 1$ and $c_1e^{c_2r^{\beta}}\le \psi(r)\le
c_3e^{c_4r^{\beta}}$ on $r>1$ for $\beta\in[0, \infty]$. A symmetric function
$\kappa(x, y)$ is bounded by two positive constants and $|\kappa(x,
y)-\kappa(x,x)|\le c_5 |x-y|^{\rho}$ for $|x-y|<1$ and $\rho>\alpha/2$. As a
corollary of our main result, we estimates sharp two-sided Green function for
this process in $C^{1,\eta}$ exterior open sets.
