Can I retrieve a key stored in a Hashset in java in some way?

I am trying to perform bi directional A star search, where in I encountered this issue.

Suppose I have an object A in a hashset H. Suppose there is another object B, such that A.equals(B) is true (and their hash values are also the same), though A and B point to different objects. Now, when I check if object B is in hashset H, it returns true, as expected. However, suppose now, I want to access some attribute in object A based on this, I then need to access object A. Note that accessing the same attribute in B will not work since they are equal only under the equals method, but do not point to the same object. How can I achieve this?

One way is to use a Hashmap, such that the value type is the same as the key type, and every time I store some key in the hashmap, I store along with it the same object as its value. But this incurs extra memory overhead of storing the value, when what I really need is a copy of the key itself. Is there any other way to achieve this?