Profile log-likelihood

Hi everyone!
I'm trying to understand the profile log-likelihood, with some trouble. To explain my trouble, I will take as an example a theoretical exercise about the profile log-likelihood of the gaussian distribution:

So in the point a, I've to compute the profile log-likelihood function in the case of n iid normal random variables. So my idea about the procedure is the following:

1)Consider the parameters of the function, select the parameter in which you are interested (in the exercise in this post, the mean of a gaussian, μ^2)

2)Compute the maximum likelihood estimation of the other parameters (in this case σ^2)

3)Return to the likelihood function, replace σ^2 with the result obtained putting the derivative of the function with respect to σ^2 and maximizing it, that's the following:
So, in the case of the exercise, I should take the likelihood function of the gaussian 1577719907553.png and replacing σ^2 with 1577719983182.png , and finally compute the logarithmic function of the result.
Is it correct? I'm not sure of this because my results are different from the ones of my classmates.