Staying on Peter And Gordon for Paul McCartney’s experiment, it seems that with the massive success of The Beatles he was beginning to wonder if he was having hits outside The Beatles just because he was in The Beatles. So he came up with a pseudonym and Bernard Webb was born.

This from Wikipedia:

Under the Lennon–McCartney moniker, McCartney had written three previous Peter and Gordon singles (viz. “A World Without Love”, “Nobody I Know”, and “I Don’t Want To See You Again”). On this occasion, McCartney used the pseudonym Bernard Webb (though some Capitol pressings carry the name A. Smith instead) to see if the song would be a success without the Lennon–McCartney credit. McCartney commented at a press conference in August 1966, “People come up to them and say, ‘Ah, we see you’re just getting in on the Lennon–McCartney bandwagon’. That’s why they did that one with our names not on it… because everyone sort of thinks that’s the reason they get hits.”However, the publishing credit was Lennon and McCartney’s company Northern Songs, and according to Gordon Waller it took only two weeks’ time for the song’s real author to be revealed, as the first review of the record said, “This Bernard Webb has an amazing talent. Could even be Paul McCartney!” It was then introduced as written by McCartney when Peter and Gordon performed it on the US TV show Hullabaloo.

Recorded in 1965 and released in February 1966, Woman went to No.14 on the US charts and No.28 in the UK.

https://en.wikipedia.org/wiki/Woman_(Paul_McCartney_song)

https://en.wikipedia.org/wiki/Paul_McCartney

https://en.wikipedia.org/wiki/Peter_and_Gordon