Salvatore Ferragamo SpA, the 90-year-old Italian fashion house famed for outfitting Hollywood stars from Marilyn Monroe to Sophia Loren, is struggling to regain its footing.

Profits are down. The brand has had difficulty in attracting younger shoppers, thanks to a minimal online presence and a product lineup that has changed little in recent years. And the company’s last chief executive unexpectedly resigned a month ago, after less than two years on the job.