A: If the advertisement does not specifically state the post is not suitable for a job share - and some headship posts do include this statement - then you should assume that in theory it is open to one.
In reality, the governors may never have considered the idea and just reacted negatively to an application. It's probably easier if you have found a job share partner, even though this is not really your responsibility.
I would approach the local authority's human resources department to ask what their policy is and how it has been communicated to schools, especially if they include a generic mention of job sharing in adverts.
Ask what you should do if you are turned down on the grounds that the governors think the post unsuitable for a job share.
You should also contact your professional association and ask whether they have had any discussions with the authorities in your area.
A different tack would be to ask the school whether the post could be split between the teaching and non-teaching components and if the non-teaching leadership role could be achieved within the fraction of the week you are seeking to work. This might actually save the school money.
John Howson is a recruitment analyst and visiting professor of education at Oxford Brookes University. To ask him a question, email him at firstname.lastname@example.org.