How to replace a character display only for the `italic` face?

Question

I am using a font that does not include italic characters. However, it does include characters like mathematical italic small a, which I would like to use instead.

I can set up a display-table that maps a regular character a to its mathematical italic counterpart 𝑎.

(let ((d (make-display-table)))
  (aset d ?a (vector ?𝑎))
  (set-window-display-table nil d))

But I can't figure out how to use this display-table only for italic text.