The Eley rounds headstamped “E C * II” are something of an enigma, but what they are not is a genuine Mark II*. These were only made in India with the exception of a small trial batch made at Royal Laboratory, Woolwich. Although the title “Mark II*” was used in some documents the correct nomenclature appears to have been “…Mark II Special”
The bullet in your Eley round appears to be similar to the Indian Dum Dum bullet with the small exposed lead tip but it is difficult to tell as the picture is out of focus. It would be interesting to pull the bullet and look at the base marking to see if it is Indian.
I suspect that these rounds were a private venture by Eley to possibly attract a military contract but have no confirmation of this. The same headstamp was used on trial short range lead bulleted rounds when the War Office was looking at training ammunition.
Attached is a photograph of my Mark II Special with a typical “D I” headstamp on the left compared to a Mark IV on the right. The exposed lead tip actually looks larger in the picture than it does in the flesh and I have another example with a smaller tip.