There is a relationship between the permittivity and the conductivity of a material, through the Kramers Kronig relations. The complex permittivity
e = e_r + j*sigma/w
where e_r is the real (normal) permittivity, sigma is the conductivity, w is the radian frequency, and j is the usual sqrt(-1). Note that e_r and sigma are both functions of frequency, and the Kramers Kronig relations show that the real and imaginary parts of a function aren't independent.
A summary is available at https://courses.cit.cornell.edu/ece533/Lectures/handout6.pdf.
Unfortunately to get the imaginary part from the real (which is what you are trying to do) you need to know the real part at all frequencies because the K-K relations involve an integral over all frequencies. Numerically you can use limited frequency data, but you still need to know the frequency variation at least a decade above and below the point of interest, and there's an arbitrary constant representing the frequency ranges left out that has to be found some other way, generally by already knowing the imaginary part at one point or having some idea about the asymptotic behavior.
Google "complex permittivity using Kramers Kronig" and you'll find several papers where people are trying to get time delay from loss or vice versa for waves in a material, which amounts to the same thing youre asking since real permittivity leads to time delay and conductivity leads to loss.