Arbitrary vs. Specific Elements (Coset)

Here is the problem I am given:

Let H be a subgroup of G and let a, b, x, y be in G. Prove or disprove:

"If a*H = b*H then a^2*H = b^2*H"

so I let the set a^2*H be {a^2 * h: h in H}

since a is in group G, a * a is in G

let a * a = c

similarly, let b * b = d in G

so we have the sets a^2*H = c*H and b^2*H = d*H

this is where my question comes into play. are a and b specific elements, such a*H = b*H does not say anything about c*H and d*H? Or can we now say c*H = d*H because of a and b.

normally i don't have any trouble with this issue, but i'm getting tripped up here for some reason.