Reflecting points on a square to make it larger

Reflecting points on a square to make it larger

You are given four points (on a Euclidian plane) that make up the corners
of a square. You may change the positions of the points by a sequence of
moves. Each move changes the position of one point, say p, to a new
location, say p', by "skipping over" one of the other 3 points. More
precisely, p skips over a point q if it is moved to the diametrically
opposed side of q. In other words, a move from p to p' is allowed if there
exists a point q such that $q = (p + p') / 2$.
Find a sequence of moves that results in a larger square. Or, if no such
sequence is possible, give a proof of why it isn't possible. (The new
square need not be oriented the same way as the original square. For
example, the larger square may be turned 45 degrees from the original, and
the larger square may have the points in a different order from in the
original square.)
I've tried many different combinations and still haven't gotten anything
more than 3 points in the right position and one point completely off. Is
this possible?