How many possible $1:1$ functions are here that satisfy the given condition?

Question

Let $A = \{ 1,2,3,4 \}$ Let $F$ be a set of all functions from $A \to A$.

Let $S$ be a relation defined by : $\forall f,g \in F$ $fSg \iff f(i) = g(i)$ for some $i \in A$

Let $h: A \to A$ be the function $h(x) = 1 $ for all $x \in A$.

How many 1:1 functions $g \in F$ are there so that $gSh$ ?

Is it $4*3*2*1$ $($total $1:1$ functions)$ - 3*2*1 (1:1$ that don't connect to $1)$ the answer?

Answer 1

It should be $4!$ (all bijections take something to $1$).

*Note a $1:1$ function from a finite set to itself is a bijection.