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How can I copy a layer into a symbol while keeping it in the center of the stage. When I copy it, the layer is of the stage and has grown.
Symbols have their own coordinate systems and transformations can be applied to their matrices on the outside.
The only way to paste something inside without changing size and position is if your symbol has its registration match that of its container and no transformations applied.
Here is an article on symbol registration which should give you better understanding on the topic.