It's a little deeper than this, software for each module is typically provided by a tier 1 or tier 2 supplier according to a spec provided by the OEM. Sometimes the tier 1 or tier 2 supplier is also subbing out the software or stuck with some system on chip that sucks.
So for a made-up example, GM wants to build a smart dash in the latest SUV, maybe Bosch or Continental has one with a SoC inside and their own software hell. OEM works with supplier to integrate, bugfix, skin, and customize. But they don't write it from scratch.
AFAIK, car manufacturers want to bring more software in house as a core competency, which is probably good because the "Tier 1"s are generally even worse at software than them and have worse aligned incentives.
So for a made-up example, GM wants to build a smart dash in the latest SUV, maybe Bosch or Continental has one with a SoC inside and their own software hell. OEM works with supplier to integrate, bugfix, skin, and customize. But they don't write it from scratch.