This is a highly interesting area that you have chosen for yourself.
Let me see if I can get my answer right.
Sai, first of all ask your respondents about the mother brand first. To be specific try to catch hold of their perceptions about the extension areas and styles of extension of the mother brand. This will cover the 'brand extension' part.
Then tell them, the reality i.e. the areas on which the mother brand is extending its brand and how. Ask the respondents their opinion about this real life brand extension. This will cover the 'brand personality' part.
Your data analysis can be based on the comparison of the perceptions of customers on brand extension and their opinion of the brand personality (after the extension has been met).
You can choose a few brands for this purpose. To come in contact with the respondents you will have to collect the primary data from where these respondents are purchasing the commodities of those brands.
Your topic is highly interesting. You can treat this as a basic or fundamental research. I don't think too many researches have been pursued on this topic. All the best to you for your research.