No InnerException in global HttpModule for ASMX Services

Question

I successfully implemented an IHttpModule for our asmx-Services like suggested in https://stackoverflow.com/a/551532/6607492

Unfortunately the serialized exception (which is in JSON-Format in our case) does not contain any information about the InnerException.

Is there any way to see the InnerException also?

My Code ist very similar to what has been given in the answer I mentioned above:

public class GlobalErrorLogModule : IHttpModule
{
    private ILogger log = Logger.GetLogger();
    private ErrorHandlerFilter errorHandlerFilter = null;

    public void Init(HttpApplication context)
    {
        context.PostRequestHandlerExecute += Context_PostRequestHandlerExecute;
        context.EndRequest += Context_EndRequest;
    }

    private void Context_PostRequestHandlerExecute(object sender, EventArgs e)
    {
        HttpApplication app = (HttpApplication)sender;
        if (app.Response.StatusCode == 500)
        {
            errorHandlerFilter = new ErrorHandlerFilter(app.Response.Filter);
            app.Response.Filter = errorHandlerFilter;
        }
    }

    private void Context_EndRequest(object sender, EventArgs e)
    {
        HttpApplication app = (HttpApplication)sender;
        if (app.Response.StatusCode == 500)
        {
            string exceptionContent = "";
            if (errorHandlerFilter != null)
            {
                exceptionContent = Encoding.UTF8.GetString(errorHandlerFilter.WrittenBytes.ToArray());
                errorHandlerFilter = null;
            }

            //The exceptionContent is in JSON-Format and does not contain information abouth the InnerException
            log.Error(exceptionContent);
        }
    }

    public void Dispose()
    {
    }
}

public class ErrorHandlerFilter : Stream
{
    private readonly Stream responseFilter;

    public List<byte> WrittenBytes { get; private set; }

    public ErrorHandlerFilter(Stream responseFilter)
    {
        this.responseFilter = responseFilter;
        WrittenBytes = new List<byte>();
    }

    public override void Write(byte[] buffer, int offset, int count)
    {
        for (int i = offset; i < offset + count; i++)
        {
            WrittenBytes.Add(buffer[i]);
        }
        responseFilter.Write(buffer, offset, count);
    }

    public override void Flush()
    {
        responseFilter.Flush();
    }

    public override long Seek(long offset, SeekOrigin origin)
    {
        return responseFilter.Seek(offset, origin);
    }

    public override void SetLength(long value)
    {
        responseFilter.SetLength(value);
    }

    public override int Read(byte[] buffer, int offset, int count)
    {
        return responseFilter.Read(buffer, offset, count);
    }

    public override bool CanRead
    {
        get { return responseFilter.CanRead; }
    }

    public override bool CanSeek
    {
        get { return responseFilter.CanSeek; }
    }

    public override bool CanWrite
    {
        get { return responseFilter.CanWrite; }
    }

    public override long Length
    {
        get { return responseFilter.Length; }
    }

    public override long Position
    {
        get { return responseFilter.Position; }
        set { responseFilter.Position = value; }
    }
}