The Girl at Central

The operator hears all!
Molly is the local telephone switchboard girl, who happens to listen in to people's calls - especially those involving Jack Reddy, whom Molly has a crush on. But Jack has eyes only for the flighty heiress Sylvia Hesketh. Sylvia agrees to elope with him, but before that can come to pass Molly hears someone else trying to cut in. Sylvia doesn't turn up for her rendezvous with Jack, and hours later is found at the side of the road over a hundred miles away, her skull caved in.
Who killed Sylvia, and why was she there on that desolate pathway?