Hhmmpph... I'm stumped.

I'd have hazzarded a guess at a pack of cheeky little Frazer Nashes, but the marque is so inextricably linked with sodomy it cannot be that. The grilles don't slope sufficiently for Lea Francis, so I can only think of Riley.
The trouble with Riley is the little known "unnatural affection" towards avifauna held by the eponymous company founder Percy Riley. The names of the cars are a giveaway of "the love that dare not squawk it's name". The Riley "Gamecock" is the most obvious hint of the deviance beheld within the company, but you could just as easily have a Riley Kestrel, Falcon or Lincock. It's not an uncommon practice, I had a friend in Sutton Coldfield many years ago, who used to get a hard-on, put a little pile of Trill on the helmet of his winkle and get his mum's budgerigar to walk up and down the shaft digging in his little claws and pecking away. Apparently it was very nice.
The Riley was one of the most successful racing cars of the late 1920s and early 1930s, particularly in hill climbs and at Le Mans, providing a platform for the success of motorsports' first women racing drivers like Kay Petrie and Dorothy Champney. It was later discovered at a Brooklands meeting in 1939 that these two ladies were in fact MEN DRESSED UP AS FLAPPERS! Indeed, Champney was discovered to have been racing with an Amazonian Macaw parrot up his/her skirt. Fortuitously, Hitler invaded Poland the next day and the whole filthy scandal was overshadowed and remained forgotten until I made it up this afternoon.
So H, I say Riley, but for a bonus point, you're wrong about the animal buggery.