Do you have a particular need for copy and paste? This can be slow and inefficient. If you're just copying data from one sheet to another, you can set the values of one range equal to the values of another range and avoid the whole thing.
Sheets("Sheet2").Range("E1:H20").Value = Sheets("Sheet1").Range("A1:D20").Value
This will set the range from cells E1:H20 on Sheet2 to the same values as those in the range A1:D20 on Sheet1, which is effectively a copy and paste. I should add that this will work only for the values themselves.
If there is specific formatting (or formulas) that you need copied and pasted, this method won't work.