 ```python
import numpy as np
import discretize
import matplotlib.pyplot as plt
from SimPEG import (
    maps,
    surveys,
    data,
    inversion,
    optimization,
    utils,
)
from SimPEG.electromagnetics.frequency_domain import FDEM
from SimPEG.meshing.simple_mesh import SimpleMesh

# Create cylindrically symmetric mesh
radius = np.logspace(-1, 1, 30)
theta = np.linspace(0, 2 * np.pi, 50)
mesh = SimpleMesh([radius, theta], radial_only=True)

# Define geologic parameters and electrical conductivity
sigma = 0.1
rho = 1 / sigma
m_rho = maps.ExpMap(np.log(rho))

# Set up relative magnetic permeability
mu_r = np.ones(mesh.nC)
mu_r[0] = 2  # Relative permeability of the central cylinder
mu_map = maps.IdentityMap(nP=mesh.nC)
mu_map *= mu_r

# Define mappings
sigma_map = maps.IdentityMap(nP=mesh.nC)

# Set up FDEM problem and survey
problem = FDEM.Problem3D_CC(
    mesh,
    sigma_map=sigma_map,
    mu_map=mu_map,
)
survey = surveys.BaseFDEMSurvey(
    [problem],
    # Add your survey parameters here
)

# Set up FDEM inversion
inversion_opts = {
    # Add your inversion options here
}
inv = inversion.Inversion(
    survey,
    problem,
    **inversion_opts
)

# Set up inversion directives
directives = {
    # Add your directive parameters here
}

# Run the inversion
if inv.finalize():
    m0 = sigma_map * rho
    inv.run(m0, directiveList=directives)

# Plot the conductivity model, the permeability model, and the data misfits
if plot_results:
    fig = plt.figure(figsize=(12, 8))
    ax1 = fig.add_subplot(121)
    problem.plot_slice(
        np.log(m_rho * rho),
        ax=ax1,
        normal='Z',
        grid=True,
        clim=[-1, 1],
    )
    ax1.set_title('Conductivity Model')

    ax2 = fig.add_subplot(122)
    problem.plot_slice(
        mu_r,
        ax=ax2,
        normal='Z',
        grid=True,
        clim=[0.5, 2.5],
    )
    ax2.set_title('Permeability Model')

    fig.tight_layout()

    # Plot data misfits
    misfits = inv.get_data_misfit()
    plt.figure()
    plt.semilogy(misfits, 'b-')
    plt.xlabel('Iteration')
    plt.ylabel('Data misfit')
    plt.title('Data misfit vs. iteration')
    plt.grid()
    plt.show()
```