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Is there any benefits to have FC layer at the middle of CNN network?

For example, in this network, FC7 has kernel size is 1. What is the benefits of using kernel size 1 in this use?

Those inception net, kernel size 1 is used for lower down the computation.

Is that same purpose or for different advantages?

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A kernel size of 1 does not make it a fully connected layer. This type of layer is called a 1x1 convolution (or "network-in-network") layer. As you noted, this is often used to reduce computational costs.

There are a few other questions and answers on this site regarding the purpose of this type of layer:

What does 1x1 convolution mean in a neural network?

how does the addition of 1×1 layers reduce from the complexity and solve sparse structure in inception module?

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