SEO Consultants are the most similar to a fractional head of SEO, in that they will often create and manage the strategy and decision making for their clients and are sometimes engaged on a fractional basis.
What differs us them from consultants is that they typically also do the work on an on-going basis, which means they must juggle the strategy/head-of work with the time they need to be hands-on.
We deliberately do not engage with clients as the people who are doing the SEO work on an on-going basis, as we are aimed at supporting business leadership in decision making and solving complicated problems. We typically and happily work alongside on-going consultants for the businesses we support, helping with increasing bandwidth or lending more experience to an issue or strategy.