Let $G$ be a finite group, and $K$ be a finite abstract $G$-simplicial complex. We say $K$ is fixed point free if for each $x$ in geometric realization of $K$, $||K||$, there exist a $g\in G$ such that $gx\neq x$. It is easy to check that if for every simplex $A\in K$ there exist a $g\in G$ such that $gA\neq A$, then $K$ is fixed point free.
I just wanted to know: Is there any necessary and sufficient conditions on $K$ for being fixed point free?
The condition you mentioned is both necessary and sufficient. To see necessity, note that if $A\in K$ and $gA=A$, then $g$ fixes the barycenter of $A$ in the geometric realization.