Let $X$ be a scheme, $S$ a $K3$ surface and $F$ a flat family of coherent sheaves on $S$ parametrized by $X$. Let us assume that for every $x\in X$ $F\_x$ is locally free, has fixed Chern classes and satisfies $h^1(F\_x)=h^2(F\_x)=0$. Does there exist a scheme $A$ together with a morphism $p:A\to X$ such that $p^{-1}(x)$ is isomorphic to $Aut(F\_x)$ for every $x\in X$? (by $F\_x$ I mean $F|\_{X\_x}$ with $X\_x$ the fiber of the projection $X\times S\to X$ over the point $x$)



Families of sheaves and automorphisms
There are standard ways of constructing this kind of objects, but I can't immediately think of a reference, so here it goes:


Let $p:Y\to X$ be a projective map (in your case, $Y=S\times X$), let $F$ and $G$ be coherent sheaves on $Y$ with $G$ flat over $X$. Then there is a scheme $H$ of finite type over $X$ whose fiber over
$x$ is the space $Hom(F\_x,G\_x)$. (More concretely, $H$ represents certain natural functor.)
In your case, $F=G$, and your $A$ is an open subscheme of $H$ consisting of invertible homomorphisms, i.e., automorphisms.


Sketch of construction of $H$: Write $F$ as the cokernel of a map 
$$d:O(-n\_1)^{N\_1}\to O(-n\_2)^{N\_2}$$
for $n\_1,n\_2\gg 0$. 
Morphisms from $O(-n)$ to $G$ correspond to sections of $p\_\*(G(n))$; by assumptions,
it is a vector bundle if $n\gg0$. Let $G\_n$ be the total space of this vector bundle. Then $d$ induces a map 
$$(G\_{n\_2})^{N\_2}\to (G\_{n\_1})^{N\_1},$$
and $H$ is the preimage of the zero section.


