Fourth moment of a symmetric square-root chi-square

Let $X$ be a random variable such that $X$ and $-X$ have the same distribution. Let $Y=X^2$ be a continuous random variable with the probability density function $$g(y)=\begin{cases}\dfrac{e^{-y/2}}{\sqrt{2\pi y}},&\text{if }y>0,\ 0,&\text{if }y\le 0.\end{cases}$$ Then $E\bigl((X-1)^4\bigr)$ is equal to