They should be responsibile for identifying opportunities to expand the company's products or services as well as being responsible for identifying any risks to current products or services and identifying ways to capitalise on the things that already are making the company money.
They should work closely with the company's product director, financial director and marketing directors.
In addition to working with the directors of these departments, they should essentially be an ear to the ground across all departments, identifying trends from internal feedback and observations.
It would have to be a business big enough for a matrix organisational setup to make sense. How would you split responsibility between the Head of Growth and sales, business development, distribution, partner relations, etc.? Only if it was a group of financial companies, or a setup in many countries, it could make sense to create a role like that - sort of like a Group Head of Pricing, who would maintain central principles, that would become operational in the more functional layers of the company.
I would recommend to have a growth head for large corporations to seriously looking into innovation to identify next sources of growth. The head of growth in large corporations required a different dynamic compared to startups in terms of scaling, the ability to execute change, maturity, systems. the key is to identify, type of growth required by the company if whether to grow aggressive, organic or just a specific new areas that you need to focus. Different type of growth required different skillset and dynamic to make it happen and ensure it is align with overall plan of a company. Specifically in financial sector, the head of growth is to do "industrial espionage" work or we called it as stalking. Stalking has been always been underrated strategy but it has proven it can be a great tool to succeed.