The tension in the string is $F$. Why? Consider the entire system as a whole. Two equal forces acting on either side, so the net force is zero and the center of mass of the system does not move. Now consider just the object A. There's a leftward force $F$ acting on the object. We now have two possibilities: A moves to the right, or A stays where it is (moving to the left is not an option because the string is inextensible).
Case 1: A stays where it is. This implies there's a force on A cancelling the leftward force $F$. Which means the force applied by the string must be $F$.
Case 2: A moves to the right. This implies the tension in the string will be more than the force you've applied. Since the string doesn't have any means to exert a force of its own on the objects, this can not be the case.
Bottom line: Tension in the string is $F$.