Inspector Lestrade thinks he has an open and shut case when Lady Beryl confesses to the killing of a man in her home. As the bullet wound was in the back of the dead man's head, it might not even be self defence.
Lady Beryl (Paulette Goddard) was born in Austria, raised in the USA and then came to live in Britain when she married Lord Beryl.
The gun used was of a German kind, more proof that this was Lady Beryl's gun according to Lestrade. Although Sherlock Holmes notes that the dead man was also Austrian and the gun could had belonged to him.
There is very little investigation in this episode. Holmes is in his study doing some experiments with poisons. He seems to have solved most of it as he talks to Lestrade and Watson.
The episode very much gives away the culprit by their shifty appearance. It really is a case of how and why.
You are left wondering if Lestrade has any ideas of basic forensics.