A function that maps Held symbol to string

Question

Rougly my question is about constructing a function that maps Hold[var] to "var"
even when var has an assigned value. In details :

In[1] var=22; A=Hold[var];

Can we construct a function F such that F[A] becomes the string, "var" ?

In fact, I can do it like :

In[2] F[x_]:=StringDrop[StringDrop[ToString[x],5],-1];
In[3] F[A]//InputForm
Out[3] "var"

But above method seems somewhat unnatural.

One natural,ideal way I think possible is :

1. uncover Hold in Hold[var], with its content(=var) untouched, and apply ToString to the content.

Another natural,ideal way I think possible is :

2. Change Hold in Hold[var] to ToString, with its content(=var) untouched.

Can you suggest natural ways to do it?