Not clear what do you mean by "assessment in school"? Do you mean students' assessment?
Quality of students' assessment in school is dependent on several factors including 1) the curriculum, and teaching and learning, 2) Student engagement and healthy environment in the classrooms, 3) using valid and reliable assessment tools, 4) giving feedback to students following formative assessment, 5) availability of resources, and funds in the schools, computers, labs etc., and then, 6) quality of teachers, and continuing professional development, and 7) effective management, student-learners relationships, and leadership in the school.
Therefore, management could be one factor among many.
All school managers, including heads and governors and others in leadership positions would argue that they are in the best position to make decisions that will enable their pupils to make good progress and that this would be reflected in their results. However, where schools have been given complete autonomy we have seen that pupils in more deprived areas are likely to have fewer opportunities to progress into more aspirational roles. However, ensuring that pupil circumstances and local requirements are adequately accounted for is a minefield. There is , therefore, a tension between ensuring appropriately challenging goals for all pupils (within a school) and ensuring that school based leaders are allowed to make the best decisions for their pupils based on local knowledge.