Not In Ric, Constantine, Sol Reverse, London Mint.

Not my area, but I wonder: could this be a mule with an obverse from a "SOLI INVICTO" type and a reverse from a "CONSTANTINVS P AVG" type?

Coins with this reverse type and inscription from London were all produced in a single series from mid-310 to late 312 A.D. The RIC listings for this series cover the range RIC VI 150-192, all for Constantine but with 30 bust types and 8 different inscriptions in various combinations.

The obverse inscription on this coin is listed as type 1b.
The obverse but type is not clear to me, but appears to be a laureate, bare bust facing right, which corresponds to bust type A. If it is cuirassed, it is type C. If draped it is B and if draped and cuirassed it is D.

However,

This obverse inscription type with the leading IMP is not listed in RIC for use with this reverse.
Of the busts, type A is not listed for this reverse. Bust types B, C, D are listed with a similar inscription but without the leading IMP.

The only use of this obverse inscription for the entire period when these reverses were being minted is paired with a r facing laureate bust draped and cuirassed, but with a CONCORD MILIT reverse.

Your coin is not listed in RIC. Unfortunately I do not know the status of any new-find updates for these coins.

That's about all I can do for you.

That's the 1 Martin , That is the only example I have seen so far but it is in Cloke & toone's new book I have been told -
The London mint of Constantius & Constantine
I have yet to get myself a copy so cant confirm it.
I was surprised that the 2 coins were not a die match, there are a few similarities but enough difference not to match.
With only a few examples known and not much information as to why it was issued I am now stuck waiting until I get my copy of the book I mentioned, hopefully there is some more info in it for me.